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Theorem List for Intuitionistic Logic Explorer - 1301-1400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtruimfal 1301 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(( ⊤ → ⊥ ) ↔ ⊥ )

Theoremfalimtru 1302 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(( ⊥ → ⊤ ) ↔ ⊤ )

Theoremfalimfal 1303 A identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(( ⊥ → ⊥ ) ↔ ⊤ )

Theoremnottru 1304 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.)
(¬ ⊤ ↔ ⊥ )

Theoremnotfal 1305 A ¬ identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(¬ ⊥ ↔ ⊤ )

Theoremtrubitru 1306 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(( ⊤ ↔ ⊤ ) ↔ ⊤ )

Theoremtrubifal 1307 A identity. (Contributed by David A. Wheeler, 23-Feb-2018.)
(( ⊤ ↔ ⊥ ) ↔ ⊥ )

Theoremfalbitru 1308 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(( ⊥ ↔ ⊤ ) ↔ ⊥ )

Theoremfalbifal 1309 A identity. (Contributed by Anthony Hart, 22-Oct-2010.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(( ⊥ ↔ ⊥ ) ↔ ⊤ )

Theoremtruxortru 1310 A identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
(( ⊤ ⊻ ⊤ ) ↔ ⊥ )

Theoremtruxorfal 1311 A identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
(( ⊤ ⊻ ⊥ ) ↔ ⊤ )

Theoremfalxortru 1312 A identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
(( ⊥ ⊻ ⊤ ) ↔ ⊤ )

Theoremfalxorfal 1313 A identity. (Contributed by David A. Wheeler, 2-Mar-2018.)
(( ⊥ ⊻ ⊥ ) ↔ ⊥ )

1.2.16  Stoic logic indemonstrables (Chrysippus of Soli)

The Greek Stoics developed a system of logic. The Stoic Chrysippus, in particular, was often considered one of the greatest logicians of antiquity. Stoic logic is different from Aristotle's system, since it focuses on propositional logic, though later thinkers did combine the systems of the Stoics with Aristotle. Jan Lukasiewicz reports, "For anybody familiar with mathematical logic it is self-evident that the Stoic dialectic is the ancient form of modern propositional logic" ( On the history of the logic of proposition by Jan Lukasiewicz (1934), translated in: Selected Works - Edited by Ludwik Borkowski - Amsterdam, North-Holland, 1970 pp. 197-217, referenced in "History of Logic" https://www.historyoflogic.com/logic-stoics.htm). For more about Aristotle's system, see barbara and related theorems.

A key part of the Stoic logic system is a set of five "indemonstrables" assigned to Chrysippus of Soli by Diogenes Laertius, though in general it is difficult to assign specific ideas to specific thinkers. The indemonstrables are described in, for example, [Lopez-Astorga] p. 11 , [Sanford] p. 39, and [Hitchcock] p. 5. These indemonstrables are modus ponendo ponens (modus ponens) ax-mp 7, modus tollendo tollens (modus tollens) mto 588, modus ponendo tollens I mptnan 1314, modus ponendo tollens II mptxor 1315, and modus tollendo ponens (exclusive-or version) mtpxor 1317. The first is an axiom, the second is already proved; in this section we prove the other three. Since we assume or prove all of indemonstrables, the system of logic we use here is as at least as strong as the set of Stoic indemonstrables. Note that modus tollendo ponens mtpxor 1317 originally used exclusive-or, but over time the name modus tollendo ponens has increasingly referred to an inclusive-or variation, which is proved in mtpor 1316. This set of indemonstrables is not the entire system of Stoic logic.

Theoremmptnan 1314 Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic. See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after mptxor 1315) as a "safer, and these days much more common" version of modus ponendo tollens because it avoids confusion between inclusive-or and exclusive-or. (Contributed by David A. Wheeler, 3-Jul-2016.)
φ    &    ¬ (φ ψ)        ¬ ψ

Theoremmptxor 1315 Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic. Note that this uses exclusive-or . See rule 2 on [Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in [Hitchcock] p. 5 . (Contributed by David A. Wheeler, 2-Mar-2018.)
φ    &   (φψ)        ¬ ψ

Theoremmtpor 1316 Modus tollendo ponens (inclusive-or version), aka disjunctive syllogism. This is similar to mtpxor 1317, one of the five original "indemonstrables" in Stoic logic. However, in Stoic logic this rule used exclusive-or, while the name modus tollendo ponens often refers to a variant of the rule that uses inclusive-or instead. The rule says, "if φ is not true, and φ or ψ (or both) are true, then ψ must be true." An alternative phrasing is, "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth." -- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four, ch. 6). (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.)
¬ φ    &   (φ ψ)       ψ

Theoremmtpxor 1317 Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, similar to mtpor 1316, one of the five "indemonstrables" in Stoic logic. The rule says, "if φ is not true, and either φ or ψ (exclusively) are true, then ψ must be true." Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtpor 1316. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mptxor 1315, that is, it is exclusive-or df-xor 1267), rule 3 of [Sanford] p. 39 (where it is not as clearly stated which kind of "or" is used but it appears to be in the same sense as mptxor 1315), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 4-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.) (Proof shortened by BJ, 19-Apr-2019.)
¬ φ    &   (φψ)       ψ

Theoremstoic2a 1318 Stoic logic Thema 2 version a.

Statement T2 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 2 as follows: "When from two assertibles a third follows, and from the third and one (or both) of the two another follows, then this other follows from the first two."

Bobzien uses constructs such as φ, ψχ; in Metamath we will represent that construct as φ ψχ.

This version a is without the phrase "or both"; see stoic2b 1319 for the version with the phrase "or both". We already have this rule as syldan 266, so here we show the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)

((φ ψ) → χ)    &   ((φ χ) → θ)       ((φ ψ) → θ)

Theoremstoic2b 1319 Stoic logic Thema 2 version b. See stoic2a 1318.

Version b is with the phrase "or both". We already have this rule as mpd3an3 1233, so here we prove the equivalence and discourage its use. (New usage is discouraged.) (Contributed by David A. Wheeler, 17-Feb-2019.)

((φ ψ) → χ)    &   ((φ ψ χ) → θ)       ((φ ψ) → θ)

Theoremstoic3 1320 Stoic logic Thema 3.

Statement T3 of [Bobzien] p. 116-117 discusses Stoic logic thema 3.

"When from two (assemblies) a third follows, and from the one that follows (i.e., the third) together with another, external external assumption, another follows, then other follows from the first two and the externally co-assumed one. (Simp. Cael. 237.2-4)" (Contributed by David A. Wheeler, 17-Feb-2019.)

((φ ψ) → χ)    &   ((χ θ) → τ)       ((φ ψ θ) → τ)

Theoremstoic4a 1321 Stoic logic Thema 4 version a.

Statement T4 of [Bobzien] p. 117 shows a reconstructed version of Stoic logic thema 4: "When from two assertibles a third follows, and from the third and one (or both) of the two and one (or more) external assertible(s) another follows, then this other follows from the first two and the external(s)."

We use θ to represent the "external" assertibles. This is version a, which is without the phrase "or both"; see stoic4b 1322 for the version with the phrase "or both". (Contributed by David A. Wheeler, 17-Feb-2019.)

((φ ψ) → χ)    &   ((χ φ θ) → τ)       ((φ ψ θ) → τ)

Theoremstoic4b 1322 Stoic logic Thema 4 version b.

This is version b, which is with the phrase "or both". See stoic4a 1321 for more information. (Contributed by David A. Wheeler, 17-Feb-2019.)

((φ ψ) → χ)    &   (((χ φ ψ) θ) → τ)       ((φ ψ θ) → τ)

1.2.17  Logical implication (continued)

Theoremsyl6an 1323 A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.)
(φψ)    &   (φ → (χθ))    &   ((ψ θ) → τ)       (φ → (χτ))

Theoremsyl10 1324 A nested syllogism inference. (Contributed by Alan Sare, 17-Jul-2011.)
(φ → (ψχ))    &   (φ → (ψ → (θτ)))    &   (χ → (τη))       (φ → (ψ → (θη)))

Theoremexbir 1325 Exportation implication also converting head from biconditional to conditional. (Contributed by Alan Sare, 31-Dec-2011.)
(((φ ψ) → (χθ)) → (φ → (ψ → (θχ))))

Theorem3impexp 1326 impexp 250 with a 3-conjunct antecedent. (Contributed by Alan Sare, 31-Dec-2011.)
(((φ ψ χ) → θ) ↔ (φ → (ψ → (χθ))))

Theorem3impexpbicom 1327 3impexp 1326 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.)
(((φ ψ χ) → (θτ)) ↔ (φ → (ψ → (χ → (τθ)))))

Theorem3impexpbicomi 1328 Deduction form of 3impexpbicom 1327. (Contributed by Alan Sare, 31-Dec-2011.)
((φ ψ χ) → (θτ))       (φ → (ψ → (χ → (τθ))))

Theoremancomsimp 1329 Closed form of ancoms 255. (Contributed by Alan Sare, 31-Dec-2011.)
(((φ ψ) → χ) ↔ ((ψ φ) → χ))

Theoremexpcomd 1330 Deduction form of expcom 109. (Contributed by Alan Sare, 22-Jul-2012.)
(φ → ((ψ χ) → θ))       (φ → (χ → (ψθ)))

Theoremexpdcom 1331 Commuted form of expd 245. (Contributed by Alan Sare, 18-Mar-2012.)
(φ → ((ψ χ) → θ))       (ψ → (χ → (φθ)))

Theoremsimplbi2comg 1332 Implication form of simplbi2com 1333. (Contributed by Alan Sare, 22-Jul-2012.)
((φ ↔ (ψ χ)) → (χ → (ψφ)))

Theoremsimplbi2com 1333 A deduction eliminating a conjunct, similar to simplbi2 367. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Wolf Lammen, 10-Nov-2012.)
(φ ↔ (ψ χ))       (χ → (ψφ))

Theoremsyl6ci 1334 A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.)
(φ → (ψχ))    &   (φθ)    &   (χ → (θτ))       (φ → (ψτ))

Theoremmpisyl 1335 A syllogism combined with a modus ponens inference. (Contributed by Alan Sare, 25-Jul-2011.)
(φψ)    &   χ    &   (ψ → (χθ))       (φθ)

1.3  Predicate calculus mostly without distinct variables

1.3.1  Universal quantifier (continued)

The universal quantifier was introduced above in wal 1241 for use by df-tru 1246. See the comments in that section. In this section, we continue with the first "real" use of it.

Axiomax-5 1336 Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.)
(x(φψ) → (xφxψ))

Axiomax-7 1337 Axiom of Quantifier Commutation. This axiom says universal quantifiers can be swapped. One of the predicate logic axioms which do not involve equality. Axiom scheme C6' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Lemma 12 of [Monk2] p. 109 and Axiom C5-3 of [Monk2] p. 113. (Contributed by NM, 5-Aug-1993.)
(xyφyxφ)

Axiomax-gen 1338 Rule of Generalization. The postulated inference rule of predicate calculus. See e.g. Rule 2 of [Hamilton] p. 74. This rule says that if something is unconditionally true, then it is true for all values of a variable. For example, if we have proved x = x, we can conclude xx = x or even yx = x. Theorem spi 1429 shows we can go the other way also: in other words we can add or remove universal quantifiers from the beginning of any theorem as required. (Contributed by NM, 5-Aug-1993.)
φ       xφ

Theoremgen2 1339 Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
φ       xyφ

Theoremmpg 1340 Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
(xφψ)    &   φ       ψ

Theoremmpgbi 1341 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
(xφψ)    &   φ       ψ

Theoremmpgbir 1342 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
(φxψ)    &   ψ       φ

Theorema7s 1343 Swap quantifiers in an antecedent. (Contributed by NM, 5-Aug-1993.)
(xyφψ)       (yxφψ)

Theoremalimi 1344 Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
(φψ)       (xφxψ)

Theorem2alimi 1345 Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(φψ)       (xyφxyψ)

Theoremalim 1346 Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 30-Mar-2008.)
(x(φψ) → (xφxψ))

Theoremal2imi 1347 Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.)
(φ → (ψχ))       (xφ → (xψxχ))

Theoremalanimi 1348 Variant of al2imi 1347 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
((φ ψ) → χ)       ((xφ xψ) → xχ)

Syntaxwnf 1349 Extend wff definition to include the not-free predicate.
wff xφ

Definitiondf-nf 1350 Define the not-free predicate for wffs. This is read "x is not free in φ". Not-free means that the value of x cannot affect the value of φ, e.g., any occurrence of x in φ is effectively bound by a "for all" or something that expands to one (such as "there exists"). In particular, substitution for a variable not free in a wff does not affect its value (sbf 1660). An example of where this is used is stdpc5 1476. See nf2 1558 for an alternative definition which does not involve nested quantifiers on the same variable.

Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition.

To be precise, our definition really means "effectively not free," because it is slightly less restrictive than the usual textbook definition for not-free (which only considers syntactic freedom). For example, x is effectively not free in the bare expression x = x, even though x would be considered free in the usual textbook definition, because the value of x in the expression x = x cannot affect the truth of the expression (and thus substitution will not change the result). (Contributed by Mario Carneiro, 11-Aug-2016.)

(Ⅎxφx(φxφ))

Theoremnfi 1351 Deduce that x is not free in φ from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
(φxφ)       xφ

Theoremhbth 1352 No variable is (effectively) free in a theorem.

This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form (φxφ) from smaller formulas of this form. These are useful for constructing hypotheses that state "x is (effectively) not free in φ." (Contributed by NM, 5-Aug-1993.)

φ       (φxφ)

Theoremnfth 1353 No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
φ       xφ

Theoremnfnth 1354 No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.)
¬ φ       xφ

Theoremnftru 1355 The true constant has no free variables. (This can also be proven in one step with nfv 1421, but this proof does not use ax-17 1419.) (Contributed by Mario Carneiro, 6-Oct-2016.)
x

Theoremalimdh 1356 Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 4-Jan-2002.)
(φxφ)    &   (φ → (ψχ))       (φ → (xψxχ))

Theoremalbi 1357 Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(x(φψ) → (xφxψ))

Theoremalrimih 1358 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
(φxφ)    &   (φψ)       (φxψ)

Theoremalbii 1359 Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
(φψ)       (xφxψ)

Theorem2albii 1360 Inference adding 2 universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
(φψ)       (xyφxyψ)

Theoremhbxfrbi 1361 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(φψ)    &   (ψxψ)       (φxφ)

Theoremnfbii 1362 Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
(φψ)       (Ⅎxφ ↔ Ⅎxψ)

Theoremnfxfr 1363 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
(φψ)    &   xψ       xφ

Theoremnfxfrd 1364 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.)
(φψ)    &   (χ → Ⅎxψ)       (χ → Ⅎxφ)

Theoremalcoms 1365 Swap quantifiers in an antecedent. (Contributed by NM, 11-May-1993.)
(xyφψ)       (yxφψ)

Theoremhbal 1366 If x is not free in φ, it is not free in yφ. (Contributed by NM, 5-Aug-1993.)
(φxφ)       (yφxyφ)

Theoremalcom 1367 Theorem 19.5 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
(xyφyxφ)

Theoremalrimdh 1368 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(φxφ)    &   (ψxψ)    &   (φ → (ψχ))       (φ → (ψxχ))

Theoremalbidh 1369 Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
(φxφ)    &   (φ → (ψχ))       (φ → (xψxχ))

Theorem19.26 1370 Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 119. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
(x(φ ψ) ↔ (xφ xψ))

Theorem19.26-2 1371 Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
(xy(φ ψ) ↔ (xyφ xyψ))

Theorem19.26-3an 1372 Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
(x(φ ψ χ) ↔ (xφ xψ xχ))

Theorem19.33 1373 Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
((xφ xψ) → x(φ ψ))

Theoremalrot3 1374 Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(xyzφyzxφ)

Theoremalrot4 1375 Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.)
(xyzwφzwxyφ)

Theoremalbiim 1376 Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
(x(φψ) ↔ (x(φψ) x(ψφ)))

Theorem2albiim 1377 Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.)
(xy(φψ) ↔ (xy(φψ) xy(ψφ)))

Theoremhband 1378 Deduction form of bound-variable hypothesis builder hban 1439. (Contributed by NM, 2-Jan-2002.)
(φ → (ψxψ))    &   (φ → (χxχ))       (φ → ((ψ χ) → x(ψ χ)))

Theoremhb3and 1379 Deduction form of bound-variable hypothesis builder hb3an 1442. (Contributed by NM, 17-Feb-2013.)
(φ → (ψxψ))    &   (φ → (χxχ))    &   (φ → (θxθ))       (φ → ((ψ χ θ) → x(ψ χ θ)))

Theoremhbald 1380 Deduction form of bound-variable hypothesis builder hbal 1366. (Contributed by NM, 2-Jan-2002.)
(φyφ)    &   (φ → (ψxψ))       (φ → (yψxyψ))

Syntaxwex 1381 Extend wff definition to include the existential quantifier ("there exists").
wff xφ

Axiomax-ie1 1382 x is bound in xφ. One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
(xφxxφ)

Axiomax-ie2 1383 Define existential quantification. xφ means "there exists at least one set x such that φ is true." One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
(x(ψxψ) → (x(φψ) ↔ (xφψ)))

Theoremhbe1 1384 x is not free in xφ. (Contributed by NM, 5-Aug-1993.)
(xφxxφ)

Theoremnfe1 1385 x is not free in xφ. (Contributed by Mario Carneiro, 11-Aug-2016.)
xxφ

Theorem19.23ht 1386 Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 1-Feb-2015.)
(x(ψxψ) → (x(φψ) ↔ (xφψ)))

Theorem19.23h 1387 Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.)
(ψxψ)       (x(φψ) ↔ (xφψ))

Theoremalnex 1388 Theorem 19.7 of [Margaris] p. 89. To read this intuitionistically, think of it as "if φ can be refuted for all x, then it is not possible to find an x for which φ holds" (and likewise for the converse). Comparing this with dfexdc 1390 illustrates that statements which look similar (to someone used to classical logic) can be different intuitionistically due to different placement of negations. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 1-Feb-2015.) (Revised by Mario Carneiro, 12-May-2015.)
(x ¬ φ ↔ ¬ xφ)

Theoremnex 1389 Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
¬ φ        ¬ xφ

Theoremdfexdc 1390 Defining xφ given decidability. It is common in classical logic to define xφ as ¬ x¬ φ but in intuitionistic logic without a decidability condition, that is only an implication not an equivalence, as seen at exalim 1391. (Contributed by Jim Kingdon, 15-Mar-2018.)
(DECID xφ → (xφ ↔ ¬ x ¬ φ))

Theoremexalim 1391 One direction of a classical definition of existential quantification. One direction of Definition of [Margaris] p. 49. For a decidable proposition, this is an equivalence, as seen as dfexdc 1390. (Contributed by Jim Kingdon, 29-Jul-2018.)
(xφ → ¬ x ¬ φ)

1.3.2  Equality predicate (continued)

The equality predicate was introduced above in wceq 1243 for use by df-tru 1246. See the comments in that section. In this section, we continue with the first "real" use of it.

Theoremweq 1392 Extend wff definition to include atomic formulas using the equality predicate.

(Instead of introducing weq 1392 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wceq 1243. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1392 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1243. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

wff x = y

Syntaxwcel 1393 Extend wff definition to include the membership connective between classes.

(The purpose of introducing wff A B here is to allow us to express i.e. "prove" the wel 1394 of predicate calculus in terms of the wceq 1243 of set theory, so that we don't "overload" the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables A and B are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus.)

wff A B

Theoremwel 1394 Extend wff definition to include atomic formulas with the epsilon (membership) predicate. This is read "x is an element of y," "x is a member of y," "x belongs to y," or "y contains x." Note: The phrase "y includes x " means "x is a subset of y;" to use it also for x y, as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT).

This syntactical construction introduces a binary non-logical predicate symbol (epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments.

(Instead of introducing wel 1394 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wcel 1393. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1394 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1393. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

wff x y

Axiomax-8 1395 Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. This is similar to, but not quite, a transitive law for equality (proved later as equtr 1595). Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105.

Axioms ax-8 1395 through ax-16 1695 are the axioms having to do with equality, substitution, and logical properties of our binary predicate (which later in set theory will mean "is a member of"). Note that all axioms except ax-16 1695 and ax-17 1419 are still valid even when x, y, and z are replaced with the same variable because they do not have any distinct variable (Metamath's \$d) restrictions. Distinct variable restrictions are required for ax-16 1695 and ax-17 1419 only. (Contributed by NM, 5-Aug-1993.)

(x = y → (x = zy = z))

Axiomax-10 1396 Axiom of Quantifier Substitution. One of the equality and substitution axioms of predicate calculus with equality. Appears as Lemma L12 in [Megill] p. 445 (p. 12 of the preprint).

The original version of this axiom was ax-10o 1604 ("o" for "old") and was replaced with this shorter ax-10 1396 in May 2008. The old axiom is proved from this one as theorem ax10o 1603. Conversely, this axiom is proved from ax-10o 1604 as theorem ax10 1605. (Contributed by NM, 5-Aug-1993.)

(x x = yy y = x)

Axiomax-11 1397 Axiom of Variable Substitution. One of the 5 equality axioms of predicate calculus. The final consequent x(x = yφ) is a way of expressing "y substituted for x in wff φ " (cf. sb6 1766). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases.

Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1708, ax11v2 1701 and ax-11o 1704. (Contributed by NM, 5-Aug-1993.)

(x = y → (yφx(x = yφ)))

Axiomax-i12 1398 Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever z is distinct from x and y, and x = y is true, then x = y quantified with z is also true. In other words, z is irrelevant to the truth of x = y. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases.

This axiom has been modified from the original ax-12 1402 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)

(z z = x (z z = y z(x = yz x = y)))

Axiomax-bndl 1399 Axiom of bundling. The general idea of this axiom is that two variables are either distinct or non-distinct. That idea could be expressed as zz = x ¬ zz = x. However, we instead choose an axiom which has many of the same consequences, but which is different with respect to a universe which contains only one object. zz = x holds if z and x are the same variable, likewise for z and y, and xz(x = yzx = y) holds if z is distinct from the others (and the universe has at least two objects).

As with other statements of the form "x is decidable (either true or false)", this does not entail the full Law of the Excluded Middle (which is the proposition that all statements are decidable), but instead merely the assertion that particular kinds of statements are decidable (or in this case, an assertion similar to decidability).

This axiom implies ax-i12 1398 as can be seen at axi12 1407. Whether ax-bndl 1399 can be proved from the remaining axioms including ax-i12 1398 is not known.

The reason we call this "bundling" is that a statement without a distinct variable constraint "bundles" together two statements, one in which the two variables are the same and one in which they are different. (Contributed by Mario Carneiro and Jim Kingdon, 14-Mar-2018.)

(z z = x (z z = y xz(x = yz x = y)))

Axiomax-4 1400 Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all x, it is true for any specific x (that would typically occur as a free variable in the wff substituted for φ). (A free variable is one that does not occur in the scope of a quantifier: x and y are both free in x = y, but only x is free in yx = y.) Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77).

Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1338. Conditional forms of the converse are given by ax-12 1402, ax-16 1695, and ax-17 1419.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from x for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1658.

(Contributed by NM, 5-Aug-1993.)

(xφφ)

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