HomeHome Intuitionistic Logic Explorer
Theorem List (p. 15 of 86)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 1401-1500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremequidqe 1401 equid 1565 with some quantification and negation without using ax-4 1376 or ax-17 1395. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)
¬ y ¬ x = x
 
Theoremax4sp1 1402 A special case of ax-4 1376 without using ax-4 1376 or ax-17 1395. (Contributed by NM, 13-Jan-2011.)
(y ¬ x = x → ¬ x = x)
 
1.3.5  Additional intuitionistic axioms
 
Axiomax-ial 1403 x is not free in xφ. Axiom 7 of 10 for intuitionistic logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
(xφxxφ)
 
Axiomax-i5r 1404 Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.)
((xφxψ) → x(xφψ))
 
1.3.6  Predicate calculus including ax-4, without distinct variables
 
Theoremspi 1405 Inference rule reversing generalization. (Contributed by NM, 5-Aug-1993.)
xφ       φ
 
Theoremsps 1406 Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)
(φψ)       (xφψ)
 
Theoremspsd 1407 Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
(φ → (ψχ))       (φ → (xψχ))
 
Theoremnfbidf 1408 An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)
xφ    &   (φ → (ψχ))       (φ → (Ⅎxψ ↔ Ⅎxχ))
 
Theoremhba1 1409 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.)
(xφxxφ)
 
Theoremnfa1 1410 x is not free in xφ. (Contributed by Mario Carneiro, 11-Aug-2016.)
xxφ
 
Theorema5i 1411 Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.)
(xφψ)       (xφxψ)
 
Theoremnfnf1 1412 x is not free in xφ. (Contributed by Mario Carneiro, 11-Aug-2016.)
xxφ
 
Theoremhbim 1413 If x is not free in φ and ψ, it is not free in (φψ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.)
(φxφ)    &   (ψxψ)       ((φψ) → x(φψ))
 
Theoremhbor 1414 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.)
(φxφ)    &   (ψxψ)       ((φ ψ) → x(φ ψ))
 
Theoremhban 1415 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.)
(φxφ)    &   (ψxψ)       ((φ ψ) → x(φ ψ))
 
Theoremhbbi 1416 If x is not free in φ and ψ, it is not free in (φψ). (Contributed by NM, 5-Aug-1993.)
(φxφ)    &   (ψxψ)       ((φψ) → x(φψ))
 
Theoremhb3or 1417 If x is not free in φ, ψ, and χ, it is not free in (φ ψ χ). (Contributed by NM, 14-Sep-2003.)
(φxφ)    &   (ψxψ)    &   (χxχ)       ((φ ψ χ) → x(φ ψ χ))
 
Theoremhb3an 1418 If x is not free in φ, ψ, and χ, it is not free in (φ ψ χ). (Contributed by NM, 14-Sep-2003.)
(φxφ)    &   (ψxψ)    &   (χxχ)       ((φ ψ χ) → x(φ ψ χ))
 
Theoremhba2 1419 Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
(yxφxyxφ)
 
Theoremhbia1 1420 Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
((xφxψ) → x(xφxψ))
 
Theorem19.3h 1421 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.)
(φxφ)       (xφφ)
 
Theorem19.3 1422 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 Mario Carneiro, 24-Sep-2016.)
xφ       (xφφ)
 
Theorem19.16 1423 Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
xφ       (x(φψ) → (φxψ))
 
Theorem19.17 1424 Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
xψ       (x(φψ) → (xφψ))
 
Theorem19.21h 1425 Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "x is not free in φ." New proofs should use 19.21 1451 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
(φxφ)       (x(φψ) ↔ (φxψ))
 
Theorem19.21bi 1426 Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(φxψ)       (φψ)
 
Theorem19.21bbi 1427 Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.)
(φxyψ)       (φψ)
 
Theorem19.27h 1428 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(ψxψ)       (x(φ ψ) ↔ (xφ ψ))
 
Theorem19.27 1429 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
xψ       (x(φ ψ) ↔ (xφ ψ))
 
Theorem19.28h 1430 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(φxφ)       (x(φ ψ) ↔ (φ xψ))
 
Theorem19.28 1431 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
xφ       (x(φ ψ) ↔ (φ xψ))
 
Theoremnfan1 1432 A closed form of nfan 1433. (Contributed by Mario Carneiro, 3-Oct-2016.)
xφ    &   (φ → Ⅎxψ)       x(φ ψ)
 
Theoremnfan 1433 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.)
xφ    &   xψ       x(φ ψ)
 
Theoremnf3an 1434 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.)
xφ    &   xψ    &   xχ       x(φ ψ χ)
 
Theoremnford 1435 If in a context x is not free in ψ and χ, it is not free in (ψ χ). (Contributed by Jim Kingdon, 29-Oct-2019.)
(φ → Ⅎxψ)    &   (φ → Ⅎxχ)       (φ → Ⅎx(ψ χ))
 
Theoremnfand 1436 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))
 
Theoremnf3and 1437 Deduction form of bound-variable hypothesis builder nf3an 1434. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)
(φ → Ⅎxψ)    &   (φ → Ⅎxχ)    &   (φ → Ⅎxθ)       (φ → Ⅎx(ψ χ θ))
 
Theoremhbim1 1438 A closed form of hbim 1413. (Contributed by NM, 5-Aug-1993.)
(φxφ)    &   (φ → (ψxψ))       ((φψ) → x(φψ))
 
Theoremnfim1 1439 A closed form of nfim 1440. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
xφ    &   (φ → Ⅎxψ)       x(φψ)
 
Theoremnfim 1440 If x is not free in φ and ψ, it is not free in (φψ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
xφ    &   xψ       x(φψ)
 
Theoremhbimd 1441 Deduction form of bound-variable hypothesis builder hbim 1413. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.)
(φxφ)    &   (φ → (ψxψ))    &   (φ → (χxχ))       (φ → ((ψχ) → x(ψχ)))
 
Theoremnfor 1442 If x is not free in φ and ψ, it is not free in (φ ψ). (Contributed by Jim Kingdon, 11-Mar-2018.)
xφ    &   xψ       x(φ ψ)
 
Theoremhbbid 1443 Deduction form of bound-variable hypothesis builder hbbi 1416. (Contributed by NM, 1-Jan-2002.)
(φxφ)    &   (φ → (ψxψ))    &   (φ → (χxχ))       (φ → ((ψχ) → x(ψχ)))
 
Theoremnfal 1444 If x is not free in φ, it is not free in yφ. (Contributed by Mario Carneiro, 11-Aug-2016.)
xφ       xyφ
 
Theoremnfnf 1445 If x is not free in φ, it is not free in yφ. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
xφ       xyφ
 
Theoremnfalt 1446 Closed form of nfal 1444. (Contributed by Jim Kingdon, 11-May-2018.)
(yxφ → Ⅎxyφ)
 
Theoremnfa2 1447 Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
xyxφ
 
Theoremnfia1 1448 Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
x(xφxψ)
 
Theorem19.21ht 1449 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (New usage is discouraged.)
(x(φxφ) → (x(φψ) ↔ (φxψ)))
 
Theorem19.21t 1450 Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.)
(Ⅎxφ → (x(φψ) ↔ (φxψ)))
 
Theorem19.21 1451 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.) (Revised by Mario Carneiro, 24-Sep-2016.)
xφ       (x(φψ) ↔ (φxψ))
 
Theoremstdpc5 1452 An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis xφ can be thought of as emulating "x is not free in φ." With this definition, 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 nfequid 1566. 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.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
xφ       (x(φψ) → (φxψ))
 
Theoremnfimd 1453 If in a context x is not free in ψ and χ, it is not free in (ψχ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
(φ → Ⅎxψ)    &   (φ → Ⅎxχ)       (φ → Ⅎx(ψχ))
 
Theoremaaanh 1454 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
(φyφ)    &   (ψxψ)       (xy(φ ψ) ↔ (xφ yψ))
 
Theoremaaan 1455 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
yφ    &   xψ       (xy(φ ψ) ↔ (xφ yψ))
 
Theoremnfbid 1456 If in a context x is not free in ψ and χ, it is not free in (ψχ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)
(φ → Ⅎxψ)    &   (φ → Ⅎxχ)       (φ → Ⅎx(ψχ))
 
Theoremnfbi 1457 If x is not free in φ and ψ, it is not free in (φψ). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
xφ    &   xψ       x(φψ)
 
1.3.7  The existential quantifier
 
Theorem19.8a 1458 If a wff is true, it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
(φxφ)
 
Theorem19.23bi 1459 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(xφψ)       (φψ)
 
Theoremexlimih 1460 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(ψxψ)    &   (φψ)       (xφψ)
 
Theoremexlimi 1461 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
xψ    &   (φψ)       (xφψ)
 
Theoremexlimd2 1462 Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1463 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.)
(φxφ)    &   (φ → (χxχ))    &   (φ → (ψχ))       (φ → (xψχ))
 
Theoremexlimdh 1463 Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.)
(φxφ)    &   (χxχ)    &   (φ → (ψχ))       (φ → (xψχ))
 
Theoremexlimd 1464 Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 18-Jun-2018.)
xφ    &   xχ    &   (φ → (ψχ))       (φ → (xψχ))
 
Theoremexlimiv 1465* Inference from Theorem 19.23 of [Margaris] p. 90.

This inference, along with our many variants is used to implement a metatheorem called "Rule C" that is given in many logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or Rule C in Hirst and Hirst's A Primer for Logic and Proof p. 59 (PDF p. 65) at http://www.mathsci.appstate.edu/~jlh/primer/hirst.pdf.

In informal proofs, the statement "Let C be an element such that..." almost always means an implicit application of Rule C.

In essence, Rule C states that if we can prove that some element x exists satisfying a wff, i.e. xφ(x) where φ(x) has x free, then we can use φ( C ) as a hypothesis for the proof where C is a new (ficticious) constant not appearing previously in the proof, nor in any axioms used, nor in the theorem to be proved. The purpose of Rule C is to get rid of the existential quantifier.

We cannot do this in Metamath directly. Instead, we use the original φ (containing x) as an antecedent for the main part of the proof. We eventually arrive at (φψ) where ψ is the theorem to be proved and does not contain x. Then we apply exlimiv 1465 to arrive at (xφψ). Finally, we separately prove xφ and detach it with modus ponens ax-mp 7 to arrive at the final theorem ψ. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 25-Jul-2012.)

(φψ)       (xφψ)
 
Theoremexim 1466 Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
(x(φψ) → (xφxψ))
 
Theoremeximi 1467 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
(φψ)       (xφxψ)
 
Theorem2eximi 1468 Inference adding 2 existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(φψ)       (xyφxyψ)
 
Theoremeximii 1469 Inference associated with eximi 1467. (Contributed by BJ, 3-Feb-2018.)
xφ    &   (φψ)       xψ
 
Theoremalinexa 1470 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
(x(φ → ¬ ψ) ↔ ¬ x(φ ψ))
 
Theoremexbi 1471 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(x(φψ) → (xφxψ))
 
Theoremexbii 1472 Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
(φψ)       (xφxψ)
 
Theorem2exbii 1473 Inference adding 2 existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
(φψ)       (xyφxyψ)
 
Theorem3exbii 1474 Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
(φψ)       (xyzφxyzψ)
 
Theoremexancom 1475 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
(x(φ ψ) ↔ x(ψ φ))
 
Theoremalrimdd 1476 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φ → Ⅎxψ)    &   (φ → (ψχ))       (φ → (ψxχ))
 
Theoremalrimd 1477 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   xψ    &   (φ → (ψχ))       (φ → (ψxχ))
 
Theoremeximdh 1478 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
(φxφ)    &   (φ → (ψχ))       (φ → (xψxχ))
 
Theoremeximd 1479 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φ → (ψχ))       (φ → (xψxχ))
 
Theoremnexd 1480 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
(φxφ)    &   (φ → ¬ ψ)       (φ → ¬ xψ)
 
Theoremexbidh 1481 Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
(φxφ)    &   (φ → (ψχ))       (φ → (xψxχ))
 
Theoremalbid 1482 Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φ → (ψχ))       (φ → (xψxχ))
 
Theoremexbid 1483 Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φ → (ψχ))       (φ → (xψxχ))
 
Theoremexsimpl 1484 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(x(φ ψ) → xφ)
 
Theoremexsimpr 1485 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(x(φ ψ) → xψ)
 
Theoremalexdc 1486 Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1512. (Contributed by Jim Kingdon, 2-Jun-2018.)
(xDECID φ → (xφ ↔ ¬ x ¬ φ))
 
Theorem19.29 1487 Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((xφ xψ) → x(φ ψ))
 
Theorem19.29r 1488 Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
((xφ xψ) → x(φ ψ))
 
Theorem19.29r2 1489 Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.)
((xyφ xyψ) → xy(φ ψ))
 
Theorem19.29x 1490 Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
((xyφ xyψ) → xy(φ ψ))
 
Theorem19.35-1 1491 Forward direction of Theorem 19.35 of [Margaris] p. 90. The converse holds for classical logic but not (for all propositions) in intuitionistic logic (Contributed by Mario Carneiro, 2-Feb-2015.)
(x(φψ) → (xφxψ))
 
Theorem19.35i 1492 Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
x(φψ)       (xφxψ)
 
Theorem19.25 1493 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
(yx(φψ) → (yxφyxψ))
 
Theorem19.30dc 1494 Theorem 19.30 of [Margaris] p. 90, with an additional decidability condition. (Contributed by Jim Kingdon, 21-Jul-2018.)
(DECID xψ → (x(φ ψ) → (xφ xψ)))
 
Theorem19.43 1495 Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
(x(φ ψ) ↔ (xφ xψ))
 
Theorem19.33b2 1496 The antecedent provides a condition implying the converse of 19.33 1349. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1497 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.)
((¬ xφ ¬ xψ) → (x(φ ψ) ↔ (xφ xψ)))
 
Theorem19.33bdc 1497 Converse of 19.33 1349 given ¬ (xφ xψ) and a decidability condition. Compare Theorem 19.33 of [Margaris] p. 90. For a version which does not require a decidability condition, see 19.33b2 1496 (Contributed by Jim Kingdon, 23-Apr-2018.)
(DECID xφ → (¬ (xφ xψ) → (x(φ ψ) ↔ (xφ xψ))))
 
Theorem19.40 1498 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(x(φ ψ) → (xφ xψ))
 
Theorem19.40-2 1499 Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(xy(φ ψ) → (xyφ xyψ))
 
Theoremexintrbi 1500 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
(x(φψ) → (xφx(φ ψ)))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8569
  Copyright terms: Public domain < Previous  Next >