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

Theorem19.40-2 1501 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 1502 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
(x(φψ) → (xφx(φ ψ)))

Theoremexintr 1503 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
(x(φψ) → (xφx(φ ψ)))

Theoremalsyl 1504 Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)
((x(φψ) x(ψχ)) → x(φχ))

Theoremhbex 1505 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 1506 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 1507 Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. (Contributed by O'Cat, 31-Mar-2008.)
(xφyφ)

Theoremi19.24 1508 Theorem 19.24 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1493, 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 1509 Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1493, 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 1510 A closed version of one direction of 19.9 1513. (Contributed by NM, 5-Aug-1993.)
(x(φxφ) → (xφφ))

Theorem19.9t 1511 A closed version of 19.9 1513. (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 1512 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 1513 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 1514 One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1488. (Contributed by Jim Kingdon, 2-Jul-2018.)
(xφ → ¬ x ¬ φ)

Theoremexnalim 1515 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 1516 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 1517 A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.)
(xy ¬ φ → ¬ xyφ)

Theoremax6blem 1518 If x is not free in φ, it is not free in ¬ φ. This theorem doesn't use ax6b 1519 compared to hbnt 1521. (Contributed by GD, 27-Jan-2018.)
(φxφ)       φx ¬ φ)

Theoremax6b 1519 Quantified Negation. Axiom C5-2 of [Monk2] p. 113.

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

xφx ¬ xφ)

Theoremhbn1 1520 x is not free in ¬ xφ. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
xφx ¬ xφ)

Theoremhbnt 1521 Closed theorem version of bound-variable hypothesis builder hbn 1522. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
(x(φxφ) → (¬ φx ¬ φ))

Theoremhbn 1522 If x is not free in φ, it is not free in ¬ φ. (Contributed by NM, 5-Aug-1993.)
(φxφ)       φx ¬ φ)

Theoremhbnd 1523 Deduction form of bound-variable hypothesis builder hbn 1522. (Contributed by NM, 3-Jan-2002.)
(φxφ)    &   (φ → (ψxψ))       (φ → (¬ ψx ¬ ψ))

Theoremnfnt 1524 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 1525 Deduction associated with nfnt 1524. (Contributed by Mario Carneiro, 24-Sep-2016.)
(φ → Ⅎxψ)       (φ → Ⅎx ¬ ψ)

Theoremnfn 1526 Inference associated with nfnt 1524. (Contributed by Mario Carneiro, 11-Aug-2016.)
xφ       x ¬ φ

Theoremnfdc 1527 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 1528 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 1529 A deduction version of one direction of 19.9 1513. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
(ψ → Ⅎxφ)       (ψ → (xφφ))

Theorem19.9hd 1530 A deduction version of one direction of 19.9 1513. This is an older variation of this theorem; new proofs should use 19.9d 1529. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
(ψxψ)    &   (ψ → (φxφ))       (ψ → (xφφ))

Theoremexcomim 1531 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
(xyφyxφ)

Theoremexcom 1532 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
(xyφyxφ)

Theorem19.12 1533 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 1534 Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
xφ       (x(φψ) → (φxψ))

Theorem19.21-2 1535 Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)
xφ    &   yφ       (xy(φψ) ↔ (φxyψ))

Theoremnf2 1536 An alternative definition of df-nf 1326, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎxφ ↔ (xφxφ))

Theoremnf3 1537 An alternative definition of df-nf 1326. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎxφx(xφφ))

Theoremnf4dc 1538 Variable x is effectively not free in φ iff φ is always true or always false, given a decidability condition. The reverse direction, nf4r 1539, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
(DECID xφ → (Ⅎxφ ↔ (xφ x ¬ φ)))

Theoremnf4r 1539 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 1538. (Contributed by Jim Kingdon, 21-Jul-2018.)
((xφ x ¬ φ) → Ⅎxφ)

Theorem19.36i 1540 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 1541 Closed form of 19.36i 1540. 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 1542 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 1543* Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
x(φψ)       (φxψ)

Theorem19.38 1544 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
((xφxψ) → x(φψ))

Theorem19.23t 1545 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 1546 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 1547 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 1548 One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if φ is decidable, as seen at 19.32dc 1547. (Contributed by Jim Kingdon, 28-Jul-2018.)
xφ       ((φ xψ) → x(φ ψ))

Theorem19.31r 1549 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 1550 Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
xψ       (x(φ ψ) ↔ (xφ ψ))

Theorem19.45 1551 Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
xφ       (x(φ ψ) ↔ (φ xψ))

Theorem19.34 1552 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
((xφ xψ) → x(φ ψ))

Theorem19.41h 1553 Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1554 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 1554 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 1555 Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1556 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)
(φxφ)       (x(φ ψ) ↔ (φ xψ))

Theorem19.42 1556 Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
xφ       (x(φ ψ) ↔ (φ xψ))

Theoremexcom13 1557 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(xyzφzyxφ)

Theoremexrot3 1558 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(xyzφyzxφ)

Theoremexrot4 1559 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
(xyzwφzwxyφ)

Theoremnexr 1560 Inference from 19.8a 1460. (Contributed by Jeff Hankins, 26-Jul-2009.)
¬ xφ        ¬ φ

Theoremexan 1561 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 1562 Deduction form of bound-variable hypothesis builder hbex 1505. (Contributed by NM, 2-Jan-2002.)
(φyφ)    &   (φ → (ψxψ))       (φ → (yψxyψ))

Theoremeeor 1563 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 1564 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 1312 through ax-14 1382 and ax-17 1396, all axioms other than ax-9 1401 are believed to be theorems of free logic, although the system without ax-9 1401 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
x x = y

Theorema9ev 1565* At least one individual exists. Weaker version of a9e 1564. (Contributed by NM, 3-Aug-2017.)
x x = y

Theoremax9o 1566 An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
(x(x = yxφ) → φ)

Theoremequid 1567 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 1568 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 1569 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1631.) 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 1570 Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)
(x = yy = x)

Theoremequcom 1571 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
(x = yy = x)

Theoremequcoms 1572 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 1573 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
(x = y → (y = zx = z))

Theoremequtrr 1574 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 1575 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 1576 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
(x = y → (x = zy = z))

Theoremequequ2 1577 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
(x = y → (z = xz = y))

Theoremelequ1 1578 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
(x = y → (x zy z))

Theoremelequ2 1579 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
(x = y → (z xz y))

Theoremax11i 1580 Inference that has ax-11 1374 (without y) as its conclusion and doesn't require ax-10 1373, ax-11 1374, or ax-12 1379 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 1581 Show that ax-10o 1582 can be derived from ax-10 1373. An open problem is whether this theorem can be derived from ax-10 1373 and the others when ax-11 1374 is replaced with ax-11o 1682. See theorem ax10 1583 for the rederivation of ax-10 1373 from ax10o 1581.

Normally, ax10o 1581 should be used rather than ax-10o 1582, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)

(x x = y → (xφyφ))

Axiomax-10o 1582 Axiom ax-10o 1582 ("o" for "old") was the original version of ax-10 1373, before it was discovered (in May 2008) that the shorter ax-10 1373 could replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of the preprint).

This axiom is redundant, as shown by theorem ax10o 1581.

Normally, ax10o 1581 should be used rather than ax-10o 1582, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

(x x = y → (xφyφ))

Theoremax10 1583 Rederivation of ax-10 1373 from original version ax-10o 1582. See theorem ax10o 1581 for the derivation of ax-10o 1582 from ax-10 1373.

This theorem should not be referenced in any proof. Instead, use ax-10 1373 above so that uses of ax-10 1373 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

(x x = yy y = x)

Theoremhbae 1584 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
(x x = yzx x = y)

Theoremnfae 1585 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
zx x = y

Theoremhbaes 1586 Rule that applies hbae 1584 to antecedent. (Contributed by NM, 5-Aug-1993.)
(zx x = yφ)       (x x = yφ)

Theoremhbnae 1587 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.)
x x = yz ¬ x x = y)

Theoremnfnae 1588 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
z ¬ x x = y

Theoremhbnaes 1589 Rule that applies hbnae 1587 to antecedent. (Contributed by NM, 5-Aug-1993.)
(z ¬ x x = yφ)       x x = yφ)

Theoremnaecoms 1590 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
x x = yφ)       y y = xφ)

Theoremequs4 1591 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.)
(x(x = yφ) → x(x = y φ))

Theoremequsalh 1592 A useful equivalence related to substitution. New proofs should use equsal 1593 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
(ψxψ)    &   (x = y → (φψ))       (x(x = yφ) ↔ ψ)

Theoremequsal 1593 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
xψ    &   (x = y → (φψ))       (x(x = yφ) ↔ ψ)

Theoremequsex 1594 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
(ψxψ)    &   (x = y → (φψ))       (x(x = y φ) ↔ ψ)

Theoremequsexd 1595 Deduction form of equsex 1594. (Contributed by Jim Kingdon, 29-Dec-2017.)
(φxφ)    &   (φ → (χxχ))    &   (φ → (x = y → (ψχ)))       (φ → (x(x = y ψ) ↔ χ))

Theoremdral1 1596 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.)
(x x = y → (φψ))       (x x = y → (xφyψ))

Theoremdral2 1597 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
(x x = y → (φψ))       (x x = y → (zφzψ))

Theoremdrex2 1598 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
(x x = y → (φψ))       (x x = y → (zφzψ))

Theoremdrnf1 1599 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
(x x = y → (φψ))       (x x = y → (Ⅎxφ ↔ Ⅎyψ))

Theoremdrnf2 1600 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
(x x = y → (φψ))       (x x = y → (Ⅎzφ ↔ Ⅎzψ))

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