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Theorem List for Intuitionistic Logic Explorer - 1501-1600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnexd 1501 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
(φxφ)    &   (φ → ¬ ψ)       (φ → ¬ xψ)

Theoremexbidh 1502 Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
(φxφ)    &   (φ → (ψχ))       (φ → (xψxχ))

Theoremalbid 1503 Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φ → (ψχ))       (φ → (xψxχ))

Theoremexbid 1504 Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
xφ    &   (φ → (ψχ))       (φ → (xψxχ))

Theoremexsimpl 1505 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 1506 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 1507 Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1533. (Contributed by Jim Kingdon, 2-Jun-2018.)
(xDECID φ → (xφ ↔ ¬ x ¬ φ))

Theorem19.29 1508 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 1509 Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
((xφ xψ) → x(φ ψ))

Theorem19.29r2 1510 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 1511 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 1512 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 1513 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 1514 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
(yx(φψ) → (yxφyxψ))

Theorem19.30dc 1515 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 1516 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 1517 The antecedent provides a condition implying the converse of 19.33 1370. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1518 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.)
((¬ xφ ¬ xψ) → (x(φ ψ) ↔ (xφ xψ)))

Theorem19.33bdc 1518 Converse of 19.33 1370 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 1517 (Contributed by Jim Kingdon, 23-Apr-2018.)
(DECID xφ → (¬ (xφ xψ) → (x(φ ψ) ↔ (xφ xψ))))

Theorem19.40 1519 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(x(φ ψ) → (xφ xψ))

Theorem19.40-2 1520 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 1521 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
(x(φψ) → (xφx(φ ψ)))

Theoremexintr 1522 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
(x(φψ) → (xφx(φ ψ)))

Theoremalsyl 1523 Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)
((x(φψ) x(ψχ)) → x(φχ))

Theoremhbex 1524 If x is not free in φ, it is not free in yφ. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
(φxφ)       (yφxyφ)

Theoremnfex 1525 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

Theorem19.2 1526 Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. (Contributed by O'Cat, 31-Mar-2008.)
(xφyφ)

Theoremi19.24 1527 Theorem 19.24 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1512, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.)
((xφxψ) → x(φψ))       ((xφxψ) → x(φψ))

Theoremi19.39 1528 Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1512, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.)
((xφxψ) → x(φψ))       ((xφxψ) → x(φψ))

Theorem19.9ht 1529 A closed version of one direction of 19.9 1532. (Contributed by NM, 5-Aug-1993.)
(x(φxφ) → (xφφ))

Theorem19.9t 1530 A closed version of 19.9 1532. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortended by Wolf Lammen, 30-Dec-2017.)
(Ⅎxφ → (xφφ))

Theorem19.9h 1531 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.)
(φxφ)       (xφφ)

Theorem19.9 1532 A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
xφ       (xφφ)

Theoremalexim 1533 One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1507. (Contributed by Jim Kingdon, 2-Jul-2018.)
(xφ → ¬ x ¬ φ)

Theoremexnalim 1534 One direction of Theorem 19.14 of [Margaris] p. 90. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)
(x ¬ φ → ¬ xφ)

Theoremexanaliim 1535 A transformation of quantifiers and logical connectives. In classical logic the converse also holds. (Contributed by Jim Kingdon, 15-Jul-2018.)
(x(φ ¬ ψ) → ¬ x(φψ))

Theoremalexnim 1536 A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.)
(xy ¬ φ → ¬ xyφ)

Theoremax6blem 1537 If x is not free in φ, it is not free in ¬ φ. This theorem doesn't use ax6b 1538 compared to hbnt 1540. (Contributed by GD, 27-Jan-2018.)
(φxφ)       φx ¬ φ)

Theoremax6b 1538 Quantified Negation. Axiom C5-2 of [Monk2] p. 113.

(Contributed by GD, 27-Jan-2018.)

xφx ¬ xφ)

Theoremhbn1 1539 x is not free in ¬ xφ. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
xφx ¬ xφ)

Theoremhbnt 1540 Closed theorem version of bound-variable hypothesis builder hbn 1541. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
(x(φxφ) → (¬ φx ¬ φ))

Theoremhbn 1541 If x is not free in φ, it is not free in ¬ φ. (Contributed by NM, 5-Aug-1993.)
(φxφ)       φx ¬ φ)

Theoremhbnd 1542 Deduction form of bound-variable hypothesis builder hbn 1541. (Contributed by NM, 3-Jan-2002.)
(φxφ)    &   (φ → (ψxψ))       (φ → (¬ ψx ¬ ψ))

Theoremnfnt 1543 If x is not free in φ, then it is not free in ¬ φ. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.)
(Ⅎxφ → Ⅎx ¬ φ)

Theoremnfnd 1544 Deduction associated with nfnt 1543. (Contributed by Mario Carneiro, 24-Sep-2016.)
(φ → Ⅎxψ)       (φ → Ⅎx ¬ ψ)

Theoremnfn 1545 Inference associated with nfnt 1543. (Contributed by Mario Carneiro, 11-Aug-2016.)
xφ       x ¬ φ

Theoremnfdc 1546 If x is not free in φ, it is not free in DECID φ. (Contributed by Jim Kingdon, 11-Mar-2018.)
xφ       xDECID φ

Theoremmodal-5 1547 The analog in our predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.)
x ¬ φx ¬ x ¬ φ)

Theorem19.9d 1548 A deduction version of one direction of 19.9 1532. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
(ψ → Ⅎxφ)       (ψ → (xφφ))

Theorem19.9hd 1549 A deduction version of one direction of 19.9 1532. This is an older variation of this theorem; new proofs should use 19.9d 1548. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
(ψxψ)    &   (ψ → (φxφ))       (ψ → (xφφ))

Theoremexcomim 1550 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
(xyφyxφ)

Theoremexcom 1551 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
(xyφyxφ)

Theorem19.12 1552 Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.)
(xyφyxφ)

Theorem19.19 1553 Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
xφ       (x(φψ) → (φxψ))

Theorem19.21-2 1554 Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)
xφ    &   yφ       (xy(φψ) ↔ (φxyψ))

Theoremnf2 1555 An alternative definition of df-nf 1347, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎxφ ↔ (xφxφ))

Theoremnf3 1556 An alternative definition of df-nf 1347. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎxφx(xφφ))

Theoremnf4dc 1557 Variable x is effectively not free in φ iff φ is always true or always false, given a decidability condition. The reverse direction, nf4r 1558, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
(DECID xφ → (Ⅎxφ ↔ (xφ x ¬ φ)))

Theoremnf4r 1558 If φ is always true or always false, then variable x is effectively not free in φ. The converse holds given a decidability condition, as seen at nf4dc 1557. (Contributed by Jim Kingdon, 21-Jul-2018.)
((xφ x ¬ φ) → Ⅎxφ)

Theorem19.36i 1559 Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
xψ    &   x(φψ)       (xφψ)

Theorem19.36-1 1560 Closed form of 19.36i 1559. One direction of Theorem 19.36 of [Margaris] p. 90. The converse holds in classical logic, but does not hold (for all propositions) in intuitionistic logic. (Contributed by Jim Kingdon, 20-Jun-2018.)
xψ       (x(φψ) → (xφψ))

Theorem19.37-1 1561 One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.)
xφ       (x(φψ) → (φxψ))

Theorem19.37aiv 1562* Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
x(φψ)       (φxψ)

Theorem19.38 1563 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
((xφxψ) → x(φψ))

Theorem19.23t 1564 Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
(Ⅎxψ → (x(φψ) ↔ (xφψ)))

Theorem19.23 1565 Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
xψ       (x(φψ) ↔ (xφψ))

Theorem19.32dc 1566 Theorem 19.32 of [Margaris] p. 90, where φ is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
xφ       (DECID φ → (x(φ ψ) ↔ (φ xψ)))

Theorem19.32r 1567 One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if φ is decidable, as seen at 19.32dc 1566. (Contributed by Jim Kingdon, 28-Jul-2018.)
xφ       ((φ xψ) → x(φ ψ))

Theorem19.31r 1568 One direction of Theorem 19.31 of [Margaris] p. 90. The converse holds in classical logic, but not intuitionistic logic. (Contributed by Jim Kingdon, 28-Jul-2018.)
xψ       ((xφ ψ) → x(φ ψ))

Theorem19.44 1569 Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
xψ       (x(φ ψ) ↔ (xφ ψ))

Theorem19.45 1570 Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
xφ       (x(φ ψ) ↔ (φ xψ))

Theorem19.34 1571 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
((xφ xψ) → x(φ ψ))

Theorem19.41h 1572 Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1573 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
(ψxψ)       (x(φ ψ) ↔ (xφ ψ))

Theorem19.41 1573 Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
xψ       (x(φ ψ) ↔ (xφ ψ))

Theorem19.42h 1574 Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1575 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)
(φxφ)       (x(φ ψ) ↔ (φ xψ))

Theorem19.42 1575 Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
xφ       (x(φ ψ) ↔ (φ xψ))

Theoremexcom13 1576 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(xyzφzyxφ)

Theoremexrot3 1577 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(xyzφyzxφ)

Theoremexrot4 1578 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
(xyzwφzwxyφ)

Theoremnexr 1579 Inference from 19.8a 1479. (Contributed by Jeff Hankins, 26-Jul-2009.)
¬ xφ        ¬ φ

Theoremexan 1580 Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(xφ ψ)       x(φ ψ)

Theoremhbexd 1581 Deduction form of bound-variable hypothesis builder hbex 1524. (Contributed by NM, 2-Jan-2002.)
(φyφ)    &   (φ → (ψxψ))       (φ → (yψxyψ))

Theoremeeor 1582 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
yφ    &   xψ       (xy(φ ψ) ↔ (xφ yψ))

1.3.8  Equality theorems without distinct variables

Theorema9e 1583 At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1333 through ax-14 1402 and ax-17 1416, all axioms other than ax-9 1421 are believed to be theorems of free logic, although the system without ax-9 1421 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
x x = y

Theorema9ev 1584* At least one individual exists. Weaker version of a9e 1583. (Contributed by NM, 3-Aug-2017.)
x x = y

Theoremax9o 1585 An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
(x(x = yxφ) → φ)

Theoremequid 1586 Identity law for equality (reflexivity). Lemma 6 of [Tarski] p. 68. This is often an axiom of equality in textbook systems, but we don't need it as an axiom since it can be proved from our other axioms.

This proof is similar to Tarski's and makes use of a dummy variable y. It also works in intuitionistic logic, unlike some other possible ways of proving this theorem. (Contributed by NM, 1-Apr-2005.)

x = x

Theoremnfequid 1587 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. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)
y x = x

Theoremstdpc6 1588 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1650.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)
x x = x

Theoremequcomi 1589 Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)
(x = yy = x)

Theoremequcom 1590 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
(x = yy = x)

Theoremequcoms 1591 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)
(x = yφ)       (y = xφ)

Theoremequtr 1592 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
(x = y → (y = zx = z))

Theoremequtrr 1593 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)
(x = y → (z = xz = y))

Theoremequtr2 1594 A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
((x = z y = z) → x = y)

Theoremequequ1 1595 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
(x = y → (x = zy = z))

Theoremequequ2 1596 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
(x = y → (z = xz = y))

Theoremelequ1 1597 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
(x = y → (x zy z))

Theoremelequ2 1598 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
(x = y → (z xz y))

Theoremax11i 1599 Inference that has ax-11 1394 (without y) as its conclusion and doesn't require ax-10 1393, ax-11 1394, or ax-12 1399 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
(x = y → (φψ))    &   (ψxψ)       (x = y → (φx(x = yφ)))

1.3.9  Axioms ax-10 and ax-11

Theoremax10o 1600 Show that ax-10o 1601 can be derived from ax-10 1393. An open problem is whether this theorem can be derived from ax-10 1393 and the others when ax-11 1394 is replaced with ax-11o 1701. See theorem ax10 1602 for the rederivation of ax-10 1393 from ax10o 1600.

Normally, ax10o 1600 should be used rather than ax-10o 1601, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)

(x x = y → (xφyφ))

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