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Theorem List for Intuitionistic Logic Explorer - 1601-1700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem19.29r 1601 Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.29r2 1602 Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))

Theorem19.29x 1603 Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))

Theorem19.35-1 1604 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) → (∀𝑥𝜑 → ∃𝑥𝜓))

Theorem19.35i 1605 Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
𝑥(𝜑𝜓)       (∀𝑥𝜑 → ∃𝑥𝜓)

Theorem19.25 1606 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
(∀𝑦𝑥(𝜑𝜓) → (∃𝑦𝑥𝜑 → ∃𝑦𝑥𝜓))

Theorem19.30dc 1607 Theorem 19.30 of [Margaris] p. 90, with an additional decidability condition. (Contributed by Jim Kingdon, 21-Jul-2018.)
(DECID𝑥𝜓 → (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)))

Theorem19.43 1608 Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))

Theorem19.33b2 1609 The antecedent provides a condition implying the converse of 19.33 1461. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1610 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.)
((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))

Theorem19.33bdc 1610 Converse of 19.33 1461 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 1609 (Contributed by Jim Kingdon, 23-Apr-2018.)
(DECID𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓))))

Theorem19.40 1611 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))

Theorem19.40-2 1612 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(𝜑𝜓) → (∃𝑥𝑦𝜑 ∧ ∃𝑥𝑦𝜓))

Theoremexintrbi 1613 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑𝜓)))

Theoremexintr 1614 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))

Theoremalsyl 1615 Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)
((∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜒)) → ∀𝑥(𝜑𝜒))

Theoremhbex 1616 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.)
(𝜑 → ∀𝑥𝜑)       (∃𝑦𝜑 → ∀𝑥𝑦𝜑)

Theoremnfex 1617 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 1618 Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. (Contributed by O'Cat, 31-Mar-2008.)
(∀𝑥𝜑 → ∃𝑦𝜑)

Theoremi19.24 1619 Theorem 19.24 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1604, 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.)
((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))       ((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theoremi19.39 1620 Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1604, 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.)
((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))       ((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.9ht 1621 A closed version of one direction of 19.9 1624. (Contributed by NM, 5-Aug-1993.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))

Theorem19.9t 1622 A closed version of 19.9 1624. (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 1623 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.)
(𝜑 → ∀𝑥𝜑)       (∃𝑥𝜑𝜑)

Theorem19.9 1624 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 1625 One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1599. (Contributed by Jim Kingdon, 2-Jul-2018.)
(∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)

Theoremexnalim 1626 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 1627 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 1628 A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.)
(∀𝑥𝑦 ¬ 𝜑 → ¬ ∃𝑥𝑦𝜑)

Theoremax6blem 1629 If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. This theorem doesn't use ax6b 1630 compared to hbnt 1632. (Contributed by GD, 27-Jan-2018.)
(𝜑 → ∀𝑥𝜑)       𝜑 → ∀𝑥 ¬ 𝜑)

Theoremax6b 1630 Quantified Negation. Axiom C5-2 of [Monk2] p. 113.

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

(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremhbn1 1631 𝑥 is not free in ¬ ∀𝑥𝜑. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremhbnt 1632 Closed theorem version of bound-variable hypothesis builder hbn 1633. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Theoremhbn 1633 If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. (Contributed by NM, 5-Aug-1993.)
(𝜑 → ∀𝑥𝜑)       𝜑 → ∀𝑥 ¬ 𝜑)

Theoremhbnd 1634 Deduction form of bound-variable hypothesis builder hbn 1633. (Contributed by NM, 3-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))

Theoremnfnt 1635 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 1636 Deduction associated with nfnt 1635. (Contributed by Mario Carneiro, 24-Sep-2016.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥 ¬ 𝜓)

Theoremnfn 1637 Inference associated with nfnt 1635. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥 ¬ 𝜑

Theoremnfdc 1638 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 1639 The analog in our predicate calculus of axiom 5 of modal logic S5. (Contributed by NM, 5-Oct-2005.)
(¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)

Theorem19.9d 1640 A deduction version of one direction of 19.9 1624. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
(𝜓 → Ⅎ𝑥𝜑)       (𝜓 → (∃𝑥𝜑𝜑))

Theorem19.9hd 1641 A deduction version of one direction of 19.9 1624. This is an older variation of this theorem; new proofs should use 19.9d 1640. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝜓 → (𝜑 → ∀𝑥𝜑))       (𝜓 → (∃𝑥𝜑𝜑))

Theoremexcomim 1642 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
(∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑)

Theoremexcom 1643 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
(∃𝑥𝑦𝜑 ↔ ∃𝑦𝑥𝜑)

Theorem19.12 1644 Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.)
(∃𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theorem19.19 1645 Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∃𝑥𝜓))

Theorem19.21-2 1646 Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)
𝑥𝜑    &   𝑦𝜑       (∀𝑥𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))

Theoremnf2 1647 An alternate definition of df-nf 1438, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

Theoremnf3 1648 An alternate definition of df-nf 1438. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑𝜑))

Theoremnf4dc 1649 Variable 𝑥 is effectively not free in 𝜑 iff 𝜑 is always true or always false, given a decidability condition. The reverse direction, nf4r 1650, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
(DECID𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)))

Theoremnf4r 1650 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 1649. (Contributed by Jim Kingdon, 21-Jul-2018.)
((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑)

Theorem19.36i 1651 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 1652 Closed form of 19.36i 1651. 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 1653 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) → (𝜑 → ∃𝑥𝜓))

Theorem19.37aiv 1654* Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
𝑥(𝜑𝜓)       (𝜑 → ∃𝑥𝜓)

Theorem19.38 1655 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))

Theorem19.23t 1656 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 1657 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 1658 Theorem 19.32 of [Margaris] p. 90, where 𝜑 is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
𝑥𝜑       (DECID 𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)))

Theorem19.32r 1659 One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if 𝜑 is decidable, as seen at 19.32dc 1658. (Contributed by Jim Kingdon, 28-Jul-2018.)
𝑥𝜑       ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))

Theorem19.31r 1660 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       ((∀𝑥𝜑𝜓) → ∀𝑥(𝜑𝜓))

Theorem19.44 1661 Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.45 1662 Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Theorem19.34 1663 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.41h 1664 Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1665 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)       (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.41 1665 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(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.42h 1666 Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1667 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Theorem19.42 1667 Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Theoremexcom13 1668 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑧𝑦𝑥𝜑)

Theoremexrot3 1669 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑦𝑧𝑥𝜑)

Theoremexrot4 1670 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑧𝑤𝑥𝑦𝜑)

Theoremnexr 1671 Inference from 19.8a 1570. (Contributed by Jeff Hankins, 26-Jul-2009.)
¬ ∃𝑥𝜑        ¬ 𝜑

Theoremexan 1672 Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∃𝑥𝜑𝜓)       𝑥(𝜑𝜓)

Theoremhbexd 1673 Deduction form of bound-variable hypothesis builder hbex 1616. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → (∃𝑦𝜓 → ∀𝑥𝑦𝜓))

Theoremeeor 1674 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))

1.3.8  Equality theorems without distinct variables

Theorema9e 1675 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 1424 through ax-14 1493 and ax-17 1507, all axioms other than ax-9 1512 are believed to be theorems of free logic, although the system without ax-9 1512 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
𝑥 𝑥 = 𝑦

Theorema9ev 1676* At least one individual exists. Weaker version of a9e 1675. (Contributed by NM, 3-Aug-2017.)
𝑥 𝑥 = 𝑦

Theoremax9o 1677 An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Theoremequid 1678 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 1679 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 1680 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1744.) 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 1681 Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremax6evr 1682* A commuted form of a9ev 1676. The naming reflects how axioms were numbered in the Metamath Proof Explorer as of 2020 (a numbering which we eventually plan to adopt here too, but until this happens everywhere only some theorems will have it). (Contributed by BJ, 7-Dec-2020.)
𝑥 𝑦 = 𝑥

Theoremequcom 1683 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremequcomd 1684 Deduction form of equcom 1683, symmetry of equality. For the versions for classes, see eqcom 2142 and eqcomd 2146. (Contributed by BJ, 6-Oct-2019.)
(𝜑𝑥 = 𝑦)       (𝜑𝑦 = 𝑥)

Theoremequcoms 1685 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 1686 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
(𝑥 = 𝑦 → (𝑦 = 𝑧𝑥 = 𝑧))

Theoremequtrr 1687 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 1688 A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)

Theoremequequ1 1689 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremequequ2 1690 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))

Theoremelequ1 1691 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theoremelequ2 1692 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Theoremax11i 1693 Inference that has ax-11 1485 (without 𝑦) as its conclusion and doesn't require ax-10 1484, ax-11 1485, or ax-12 1490 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))    &   (𝜓 → ∀𝑥𝜓)       (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

1.3.9  Axioms ax-10 and ax-11

Theoremax10o 1694 Show that ax-10o 1695 can be derived from ax-10 1484. An open problem is whether this theorem can be derived from ax-10 1484 and the others when ax-11 1485 is replaced with ax-11o 1796. See theorem ax10 1696 for the rederivation of ax-10 1484 from ax10o 1694.

Normally, ax10o 1694 should be used rather than ax-10o 1695, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.)

(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Axiomax-10o 1695 Axiom ax-10o 1695 ("o" for "old") was the original version of ax-10 1484, before it was discovered (in May 2008) that the shorter ax-10 1484 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 1694.

Normally, ax10o 1694 should be used rather than ax-10o 1695, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑))

Theoremax10 1696 Rederivation of ax-10 1484 from original version ax-10o 1695. See theorem ax10o 1694 for the derivation of ax-10o 1695 from ax-10 1484.

This theorem should not be referenced in any proof. Instead, use ax-10 1484 above so that uses of ax-10 1484 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)

Theoremhbae 1697 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)

Theoremnfae 1698 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑧𝑥 𝑥 = 𝑦

Theoremhbaes 1699 Rule that applies hbae 1697 to antecedent. (Contributed by NM, 5-Aug-1993.)
(∀𝑧𝑥 𝑥 = 𝑦𝜑)       (∀𝑥 𝑥 = 𝑦𝜑)

Theoremhbnae 1700 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 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦)

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