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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | 19.26OLD 1301 | Obsolete proof of 19.26 1300 as of 4-Jul-2014. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀x(φ ∧ ψ) ↔ (∀xφ ∧ ∀xψ)) | ||
Theorem | 19.26-2 1302 | Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.) |
⊢ (∀x∀y(φ ∧ ψ) ↔ (∀x∀yφ ∧ ∀x∀yψ)) | ||
Theorem | 19.26-3an 1303 | Theorem 19.26 of [Margaris] p. 90 with triple conjunction. (Contributed by NM, 13-Sep-2011.) |
⊢ (∀x(φ ∧ ψ ∧ χ) ↔ (∀xφ ∧ ∀xψ ∧ ∀xχ)) | ||
Theorem | 19.33 1304 | Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ ((∀xφ ∨ ∀xψ) → ∀x(φ ∨ ψ)) | ||
Theorem | alrot3 1305 | Theorem *11.21 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∀x∀y∀zφ ↔ ∀y∀z∀xφ) | ||
Theorem | alrot4 1306 | Rotate 4 universal quantifiers twice. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 28-Jun-2014.) |
⊢ (∀x∀y∀z∀wφ ↔ ∀z∀w∀x∀yφ) | ||
Theorem | alrot4OLD 1307 | Obsolete proof of alrot4 1306 as of 28-Jun-2014. (Contributed by NM, 2-Feb-2005.) |
⊢ (∀x∀y∀z∀wφ ↔ ∀z∀w∀x∀yφ) | ||
Theorem | albiim 1308 | Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.) |
⊢ (∀x(φ ↔ ψ) ↔ (∀x(φ → ψ) ∧ ∀x(ψ → φ))) | ||
Theorem | 2albiim 1309 | Split a biconditional and distribute 2 quantifiers. (Contributed by NM, 3-Feb-2005.) |
⊢ (∀x∀y(φ ↔ ψ) ↔ (∀x∀y(φ → ψ) ∧ ∀x∀y(ψ → φ))) | ||
Theorem | hband 1310 | Deduction form of bound-variable hypothesis builder hban 1370. (Contributed by NM, 2-Jan-2002.) |
⊢ (φ → (ψ → ∀xψ)) & ⊢ (φ → (χ → ∀xχ)) ⇒ ⊢ (φ → ((ψ ∧ χ) → ∀x(ψ ∧ χ))) | ||
Theorem | hb3and 1311 | Deduction form of bound-variable hypothesis builder hb3an 1373. (Contributed by NM, 17-Feb-2013.) |
⊢ (φ → (ψ → ∀xψ)) & ⊢ (φ → (χ → ∀xχ)) & ⊢ (φ → (θ → ∀xθ)) ⇒ ⊢ (φ → ((ψ ∧ χ ∧ θ) → ∀x(ψ ∧ χ ∧ θ))) | ||
Theorem | hbald 1312 | Deduction form of bound-variable hypothesis builder hbal 1296. (Contributed by NM, 2-Jan-2002.) |
⊢ (φ → ∀yφ) & ⊢ (φ → (ψ → ∀xψ)) ⇒ ⊢ (φ → (∀yψ → ∀x∀yψ)) | ||
Syntax | wex 1313 | Extend wff definition to include the existential quantifier ("there exists"). |
wff ∃xφ | ||
Axiom | ax-ie1 1314 | x is bound in ∃xφ. Axiom 9 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ (∃xφ → ∀x∃xφ) | ||
Axiom | ax-ie2 1315 | Define existential quantification. ∃xφ means "there exists at least one set x such that φ is true." Axiom 10 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ))) | ||
Theorem | hbe1 1316 | x is not free in ∃xφ. (Contributed by NM, 5-Aug-1993.) |
⊢ (∃xφ → ∀x∃xφ) | ||
Theorem | nfe1 1317 | x is not free in ∃xφ. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎx∃xφ | ||
Theorem | 19.23t 1318 | Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 1-Feb-2015.) |
⊢ (∀x(ψ → ∀xψ) → (∀x(φ → ψ) ↔ (∃xφ → ψ))) | ||
Theorem | 19.23 1319 | Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.) |
⊢ (ψ → ∀xψ) ⇒ ⊢ (∀x(φ → ψ) ↔ (∃xφ → ψ)) | ||
Theorem | alnex 1320 | 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 df-ex 1758 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) | ||
Theorem | nex 1321 | Generalization rule for negated wff. (Contributed by NM, 18-May-1994.) |
⊢ ¬ φ ⇒ ⊢ ¬ ∃xφ | ||
Theorem | dfexdc 1322 | Defining ∃xφ given decidability. It is common in classical logic to define ∃xφ as ¬ ∀x¬ φ but in intuitionistic logic, that definition only holds under certain conditions. (Contributed by Jim Kingdon, 15-Mar-2018.) |
⊢ (DECID ∃xφ → (∃xφ ↔ ¬ ∀x ¬ φ)) | ||
Syntax | cv 1323 |
This syntax construction states that a variable x, which has been
declared to be a set variable by $f statement vx, is also a class
expression. See comments in set.mm for justification.
While it is tempting and perhaps occasionally useful to view cv 1323 as a "type conversion" from a set variable to a class variable, keep in mind that cv 1323 is intrinsically no different from any other class-building syntax. (The description above applies to set theory, not predicate calculus. The purpose of introducing class x here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1325 of predicate calculus from the wceq 1324 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.) |
class x | ||
Syntax | wceq 1324 |
Extend wff definition to include class equality.
(The purpose of introducing wff A = B here, and not in set theory where it belongs, is to allow us to express i.e. "prove" the weq 1325 of predicate calculus in terms of the wceq 1324 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. For example, some parsers - although not the Metamath program - stumble on the fact that the = in x = y could be the = of either weq 1325 or wceq 1324, although mathematically it makes no difference. 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 | ||
Theorem | weq 1325 |
Extend wff definition to include atomic formulas using the equality
predicate.
(Instead of introducing weq 1325 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 1324. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1325 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1324. 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 | ||
Syntax | wcel 1326 |
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 1327 of predicate calculus in terms of the wceq 1324 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 | ||
Theorem | wel 1327 |
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 1327 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 1326. This lets us avoid overloading the ∈ connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1327 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1326. 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 | ||
Axiom | ax-8 1328 |
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 1480). Axiom scheme C8'
in [Megill] p. 448 (p. 16 of the preprint).
Also appears as Axiom C7 of
[Monk2] p. 105.
Axioms ax-8 1328 through ax-16 1570 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 1570 and ax-17 1349 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 1570 and ax-17 1349 only. (Contributed by NM, 5-Aug-1993.) |
⊢ (x = y → (x = z → y = z)) | ||
Axiom | ax-10 1329 |
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 1489 ("o" for "old") and was replaced with this shorter ax-10 1329 in May 2008. The old axiom is proved from this one as theorem ax10o 1488. Conversely, this axiom is proved from ax-10o 1489 as theorem ax10 1490. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀x x = y → ∀y y = x) | ||
Axiom | ax-11 1330 |
Axiom of Variable Substitution. One of the 5 equality axioms of predicate
calculus. The final consequent ∀x(x = y →
φ) is a way of
expressing "y
substituted for x in wff
φ " (cf. sb6 1637).
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.
In classical logic, ax-11o 1579 can be derived from this axiom, as can be seen at ax11o 1578. However, the current proof of ax11o 1578 is not valid intuitionistically. In classical logic, this axiom is metalogically independent from the others, but not logically independent. Lack of logical independence means that if the wff expression substituted for φ contains no wff variables, the resulting statement can be proved without invoking this axiom. The current proofs of this are not valid in intuitionistic logic, however. Specifically, we can prove any wff-variable-free instance of axiom ax-11o 1579 (from which the ax-11 1330 instance follows by theorem ax11 1918.) The proof is by induction on formula length, using ax11eq 1920 and ax11el 1921 for the basis steps and ax11indn 1922, ax11indi 1923, and ax11inda 1927 for the induction steps. Many of those theorems rely on classical logic for their proofs. Variants of this axiom which are equivalent in classical logic but which have not been shown to be equivalent for intuitionistic logic are ax11v 1583, ax11v2 1576 and ax-11o 1579. (Contributed by NM, 5-Aug-1993.) |
⊢ (x = y → (∀yφ → ∀x(x = y → φ))) | ||
Axiom | ax-i12 1331 |
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 1335 for compatibility with intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y))) | ||
Axiom | ax-bnd 1332 |
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 ∀x∀z(x = y → ∀zx = 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 is similar to ax-i12 1331, but appears to be stronger. At least for now, we keep them both as distinct axioms, but they serve similar purposes. 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 ∨ ∀x∀z(x = y → ∀z x = y))) | ||
Axiom | ax-4 1333 |
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 1269. Conditional forms of the converse are given by ax-12 1335, ax-15 1917, ax-16 1570, and ax-17 1349. 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 1537. The relationship of this axiom to other predicate logic axioms is different than in the classical case. In particular, the current proof of ax4 1915 (which derives ax-4 1333 from certain other axioms) relies on ax-3 719 and so is not valid intuitionistically. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀xφ → φ) | ||
Theorem | sp 1334 | Specialization. Another name for ax-4 1333. (Contributed by NM, 21-May-2008.) |
⊢ (∀xφ → φ) | ||
Theorem | ax-12 1335 | Rederive the original version of the axiom from ax-i12 1331. Note that we need ax-4 1333 for the derivation, but the proof of ax4 1915 is nontheless non-circular since it does not use ax-12. (Contributed by Mario Carneiro, 3-Feb-2015.) |
⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y))) | ||
Theorem | ax12or 1336 | Derive the intuitionistic form of ax-12 1335 from the original form. (Contributed by NM, 3-Feb-2015.) |
⊢ (∀z z = x ∨ (∀z z = y ∨ ∀z(x = y → ∀z x = y))) | ||
Axiom | ax-13 1337 | Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of the ∈ binary predicate. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be. (Contributed by NM, 5-Aug-1993.) |
⊢ (x = y → (x ∈ z → y ∈ z)) | ||
Axiom | ax-14 1338 | Axiom of Equality. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of the ∈ binary predicate. Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. (Contributed by NM, 5-Aug-1993.) |
⊢ (x = y → (z ∈ x → z ∈ y)) | ||
Theorem | hbequid 1339 | Bound-variable hypothesis builder for x = x. This theorem tells us that any variable, including x, is effectively not free in x = x, even though x is technically free according to the traditional definition of free variable. (The proof uses only ax-5 1267, ax-8 1328, ax-12 1335, and ax-gen 1269. This shows that this can be proved without ax-9 1354, even though the theorem equid 1475 cannot be. A shorter proof using ax-9 1354 is obtainable from equid 1475 and hbth 1283.) (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 23-Mar-2014.) |
⊢ (x = x → ∀y x = x) | ||
Theorem | alequcom 1340 | Commutation law for identical variable specifiers. The antecedent and consequent are true when x and y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.) |
⊢ (∀x x = y → ∀y y = x) | ||
Theorem | alequcoms 1341 | A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀x x = y → φ) ⇒ ⊢ (∀y y = x → φ) | ||
Theorem | nalequcoms 1342 | A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 2-Feb-2015.) |
⊢ (¬ ∀x x = y → φ) ⇒ ⊢ (¬ ∀y y = x → φ) | ||
Theorem | nfr 1343 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 26-Sep-2016.) |
⊢ (Ⅎxφ → (φ → ∀xφ)) | ||
Theorem | nfri 1344 | Consequence of the definition of not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎxφ ⇒ ⊢ (φ → ∀xφ) | ||
Theorem | nfrd 1345 | Consequence of the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ (φ → Ⅎxψ) ⇒ ⊢ (φ → (ψ → ∀xψ)) | ||
Theorem | nfd 1346 | Deduce that x is not free in ψ in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ Ⅎxφ & ⊢ (φ → (ψ → ∀xψ)) ⇒ ⊢ (φ → Ⅎxψ) | ||
Theorem | nfdh 1347 | Deduce that x is not free in ψ in a context. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ (φ → ∀xφ) & ⊢ (φ → (ψ → ∀xψ)) ⇒ ⊢ (φ → Ⅎxψ) | ||
Theorem | nfrimi 1348 | Moving an antecedent outside Ⅎ. (Contributed by Jim Kingdon, 23-Mar-2018.) |
⊢ Ⅎxφ & ⊢ Ⅎx(φ → ψ) ⇒ ⊢ (φ → Ⅎxψ) | ||
Axiom | ax-17 1349* |
Axiom to quantify a variable over a formula in which it does not occur.
Axiom C5 in [Megill] p. 444 (p. 11 of the
preprint). Also appears as
Axiom B6 (p. 75) of system S2 of [Tarski]
p. 77 and Axiom C5-1 of
[Monk2] p. 113.
This axiom is logically redundant in the (logically complete) predicate calculus axiom system consisting of ax-gen 1269, ax-4 1333 through ax-9 1354, ax-10o 1489, and ax-12 1335 through ax-16 1570: in that system, we can derive any instance of ax-17 1349 not containing wff variables by induction on formula length, using ax17eq 1938 and ax17el 1940 for the basis together hbn 1445, hbal 1296, and hbim 1368. However, if we omit this axiom, our development would be quite inconvenient since we could work only with specific instances of wffs containing no wff variables - this axiom introduces the concept of a set variable not occurring in a wff (as opposed to just two set variables being distinct). (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ∀xφ) | ||
Theorem | a17d 1350* | ax-17 1349 with antecedent. (Contributed by NM, 1-Mar-2013.) |
⊢ (φ → (ψ → ∀xψ)) | ||
Theorem | nfv 1351* | If x is not present in φ, then x is not free in φ. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎxφ | ||
Theorem | nfvd 1352* | nfv 1351 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1401. (Contributed by Mario Carneiro, 6-Oct-2016.) |
⊢ (φ → Ⅎxψ) | ||
Axiom | ax-i9 1353 | Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. One thing this axiom tells us is that at least one thing exists (although ax-4 1333 and possibly others also tell us that, i.e. they are not valid in the empty domain of a "free logic"). In this form (not requiring that x and y be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) Another name for this theorem is a9e 1473, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ ∃x x = y | ||
Theorem | ax-9 1354 | Derive ax-9 1354 from ax-i9 1353, the modified version for intuitionistic logic. Although ax-9 1354 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1353. (Contributed by NM, 3-Feb-2015.) |
⊢ ¬ ∀x ¬ x = y | ||
Theorem | equidqe 1355 | equid 1475 with some quantification and negation without using ax-4 1333 or ax-17 1349. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.) |
⊢ ¬ ∀y ¬ x = x | ||
Theorem | equidqeOLD 1356 | Obsolete proof of equidqe 1355 as of 27-Feb-2014. (Contributed by NM, 13-Jan-2011.) |
⊢ ¬ ∀y ¬ x = x | ||
Theorem | ax4sp1 1357 | A special case of ax-4 1333 without using ax-4 1333 or ax-17 1349. (Contributed by NM, 13-Jan-2011.) |
⊢ (∀y ¬ x = x → ¬ x = x) | ||
Axiom | ax-ial 1358 | x is not free in ∀xφ. Axiom 7 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ (∀xφ → ∀x∀xφ) | ||
Axiom | ax-i5r 1359 | The converse of ax-5o 1913. Axiom 8 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.) |
⊢ ((∀xφ → ∀xψ) → ∀x(∀xφ → ψ)) | ||
Theorem | a4i 1360 | Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.) |
⊢ ∀xφ ⇒ ⊢ φ | ||
Theorem | a4s 1361 | Generalization of antecedent. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ψ) ⇒ ⊢ (∀xφ → ψ) | ||
Theorem | a4sd 1362 | Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.) |
⊢ (φ → (ψ → χ)) ⇒ ⊢ (φ → (∀xψ → χ)) | ||
Theorem | nfbidf 1363 | An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.) |
⊢ Ⅎxφ & ⊢ (φ → (ψ ↔ χ)) ⇒ ⊢ (φ → (Ⅎxψ ↔ Ⅎxχ)) | ||
Theorem | hba1 1364 | x is not free in ∀xφ. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀xφ → ∀x∀xφ) | ||
Theorem | nfa1 1365 | x is not free in ∀xφ. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎx∀xφ | ||
Theorem | a5i 1366 | Inference version of ax-5o 1913. (Contributed by NM, 5-Aug-1993.) |
⊢ (∀xφ → ψ) ⇒ ⊢ (∀xφ → ∀xψ) | ||
Theorem | nfnf1 1367 | x is not free in Ⅎxφ. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ ℲxℲxφ | ||
Theorem | hbim 1368 | If x is not free in φ and ψ, it is not free in (φ → ψ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.) |
⊢ (φ → ∀xφ) & ⊢ (ψ → ∀xψ) ⇒ ⊢ ((φ → ψ) → ∀x(φ → ψ)) | ||
Theorem | hbor 1369 | If x is not free in φ and ψ, it is not free in (φ ∨ ψ). (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.) |
⊢ (φ → ∀xφ) & ⊢ (ψ → ∀xψ) ⇒ ⊢ ((φ ∨ ψ) → ∀x(φ ∨ ψ)) | ||
Theorem | hban 1370 | If x is not free in φ and ψ, it is not free in (φ ∧ ψ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.) |
⊢ (φ → ∀xφ) & ⊢ (ψ → ∀xψ) ⇒ ⊢ ((φ ∧ ψ) → ∀x(φ ∧ ψ)) | ||
Theorem | hbbi 1371 | If x is not free in φ and ψ, it is not free in (φ ↔ ψ). (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ∀xφ) & ⊢ (ψ → ∀xψ) ⇒ ⊢ ((φ ↔ ψ) → ∀x(φ ↔ ψ)) | ||
Theorem | hb3or 1372 | If x is not free in φ, ψ, and χ, it is not free in (φ ∨ ψ ∨ χ). (Contributed by NM, 14-Sep-2003.) |
⊢ (φ → ∀xφ) & ⊢ (ψ → ∀xψ) & ⊢ (χ → ∀xχ) ⇒ ⊢ ((φ ∨ ψ ∨ χ) → ∀x(φ ∨ ψ ∨ χ)) | ||
Theorem | hb3an 1373 | If x is not free in φ, ψ, and χ, it is not free in (φ ∧ ψ ∧ χ). (Contributed by NM, 14-Sep-2003.) |
⊢ (φ → ∀xφ) & ⊢ (ψ → ∀xψ) & ⊢ (χ → ∀xχ) ⇒ ⊢ ((φ ∧ ψ ∧ χ) → ∀x(φ ∧ ψ ∧ χ)) | ||
Theorem | hba2 1374 | Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |
⊢ (∀y∀xφ → ∀x∀y∀xφ) | ||
Theorem | hbia1 1375 | Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.) |
⊢ ((∀xφ → ∀xψ) → ∀x(∀xφ → ∀xψ)) | ||
Theorem | 19.3 1376 | A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-May-2007.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (∀xφ ↔ φ) | ||
Theorem | 19.16 1377 | Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (∀x(φ ↔ ψ) → (φ ↔ ∀xψ)) | ||
Theorem | 19.17 1378 | Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (ψ → ∀xψ) ⇒ ⊢ (∀x(φ ↔ ψ) → (∀xφ ↔ ψ)) | ||
Theorem | 19.21 1379 | Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "x is not free in φ." (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (∀x(φ → ψ) ↔ (φ → ∀xψ)) | ||
Theorem | 19.21-2 1380 | Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.) |
⊢ (φ → ∀xφ) & ⊢ (φ → ∀yφ) ⇒ ⊢ (∀x∀y(φ → ψ) ↔ (φ → ∀x∀yψ)) | ||
Theorem | stdpc5 1381 | An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis (φ → ∀xφ) can be thought of as emulating "x is not free in φ." With this convention, the meaning of "not free" is less restrictive than the usual textbook definition; for example x would not (for us) be free in x = x by hbequid 1339. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. (Contributed by NM, 22-Sep-1993.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (∀x(φ → ψ) → (φ → ∀xψ)) | ||
Theorem | 19.21bi 1382 | Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ∀xψ) ⇒ ⊢ (φ → ψ) | ||
Theorem | 19.21bbi 1383 | Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.) |
⊢ (φ → ∀x∀yψ) ⇒ ⊢ (φ → ψ) | ||
Theorem | 19.27 1384 | Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (ψ → ∀xψ) ⇒ ⊢ (∀x(φ ∧ ψ) ↔ (∀xφ ∧ ψ)) | ||
Theorem | 19.28 1385 | Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ∀xφ) ⇒ ⊢ (∀x(φ ∧ ψ) ↔ (φ ∧ ∀xψ)) | ||
Theorem | nfan1 1386 | A closed form of nfan 1387. (Contributed by Mario Carneiro, 3-Oct-2016.) |
⊢ Ⅎxφ & ⊢ (φ → Ⅎxψ) ⇒ ⊢ Ⅎx(φ ∧ ψ) | ||
Theorem | nfan 1387 | If x is not free in φ and ψ, it is not free in (φ ∧ ψ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) |
⊢ Ⅎxφ & ⊢ Ⅎxψ ⇒ ⊢ Ⅎx(φ ∧ ψ) | ||
Theorem | nf3an 1388 | If x is not free in φ, ψ, and χ, it is not free in (φ ∧ ψ ∧ χ). (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎxφ & ⊢ Ⅎxψ & ⊢ Ⅎxχ ⇒ ⊢ Ⅎx(φ ∧ ψ ∧ χ) | ||
Theorem | nfand 1389 | If in a context x is not free in ψ and χ, it is not free in (ψ ∧ χ). (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (φ → Ⅎxψ) & ⊢ (φ → Ⅎxχ) ⇒ ⊢ (φ → Ⅎx(ψ ∧ χ)) | ||
Theorem | nf3and 1390 | Deduction form of bound-variable hypothesis builder nf3an 1388. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.) |
⊢ (φ → Ⅎxψ) & ⊢ (φ → Ⅎxχ) & ⊢ (φ → Ⅎxθ) ⇒ ⊢ (φ → Ⅎx(ψ ∧ χ ∧ θ)) | ||
Theorem | hbim1 1391 | A closed form of hbim 1368. (Contributed by NM, 5-Aug-1993.) |
⊢ (φ → ∀xφ) & ⊢ (φ → (ψ → ∀xψ)) ⇒ ⊢ ((φ → ψ) → ∀x(φ → ψ)) | ||
Theorem | nfim1 1392 | A closed form of nfim 1393. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
⊢ Ⅎxφ & ⊢ (φ → Ⅎxψ) ⇒ ⊢ Ⅎx(φ → ψ) | ||
Theorem | nfim 1393 | If x is not free in φ and ψ, it is not free in (φ → ψ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
⊢ Ⅎxφ & ⊢ Ⅎxψ ⇒ ⊢ Ⅎx(φ → ψ) | ||
Theorem | hbimd 1394 | Deduction form of bound-variable hypothesis builder hbim 1368. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.) |
⊢ (φ → ∀xφ) & ⊢ (φ → (ψ → ∀xψ)) & ⊢ (φ → (χ → ∀xχ)) ⇒ ⊢ (φ → ((ψ → χ) → ∀x(ψ → χ))) | ||
Theorem | nfor 1395 | If x is not free in φ and ψ, it is not free in (φ ∨ ψ). (Contributed by Jim Kingdon, 11-Mar-2018.) |
⊢ Ⅎxφ & ⊢ Ⅎxψ ⇒ ⊢ Ⅎx(φ ∨ ψ) | ||
Theorem | hbbid 1396 | Deduction form of bound-variable hypothesis builder hbbi 1371. (Contributed by NM, 1-Jan-2002.) |
⊢ (φ → ∀xφ) & ⊢ (φ → (ψ → ∀xψ)) & ⊢ (φ → (χ → ∀xχ)) ⇒ ⊢ (φ → ((ψ ↔ χ) → ∀x(ψ ↔ χ))) | ||
Theorem | nfal 1397 | If x is not free in φ, it is not free in ∀yφ. (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎxφ ⇒ ⊢ Ⅎx∀yφ | ||
Theorem | nfnf 1398 | If x is not free in φ, it is not free in Ⅎyφ. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |
⊢ Ⅎxφ ⇒ ⊢ ℲxℲyφ | ||
Theorem | 19.21ht 1399 | Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) |
⊢ (∀x(φ → ∀xφ) → (∀x(φ → ψ) ↔ (φ → ∀xψ))) | ||
Theorem | 19.21t 1400 | Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) |
⊢ (Ⅎxφ → (∀x(φ → ψ) ↔ (φ → ∀xψ))) |
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