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Theorem List for Intuitionistic Logic Explorer - 1601-1700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexancom 1601 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
(∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
 
Theoremalrimdd 1602 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))
 
Theoremalrimd 1603 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   𝑥𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))
 
Theoremeximdh 1604 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
Theoremeximd 1605 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
Theoremnexd 1606 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)
 
Theoremexbidh 1607 Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 
Theoremalbid 1608 Formula-building rule for universal quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
 
Theoremexbid 1609 Formula-building rule for existential quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 
Theoremexsimpl 1610 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(∃𝑥(𝜑𝜓) → ∃𝑥𝜑)
 
Theoremexsimpr 1611 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(∃𝑥(𝜑𝜓) → ∃𝑥𝜓)
 
Theoremalexdc 1612 Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1638. (Contributed by Jim Kingdon, 2-Jun-2018.)
(∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑))
 
Theorem19.29 1613 Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.29r 1614 Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.29r2 1615 Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))
 
Theorem19.29x 1616 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 1617 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 1618 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 1619 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 1620 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 1621 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 1622 The antecedent provides a condition implying the converse of 19.33 1477. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1623 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.)
((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
 
Theorem19.33bdc 1623 Converse of 19.33 1477 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 1622 (Contributed by Jim Kingdon, 23-Apr-2018.)
(DECID𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓))))
 
Theorem19.40 1624 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
 
Theorem19.40-2 1625 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 1626 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑𝜓)))
 
Theoremexintr 1627 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
 
Theoremalsyl 1628 Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)
((∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜒)) → ∀𝑥(𝜑𝜒))
 
Theoremhbex 1629 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 1630 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 1631 Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. (Contributed by O'Cat, 31-Mar-2008.)
(∀𝑥𝜑 → ∃𝑦𝜑)
 
Theoremi19.24 1632 Theorem 19.24 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1617, 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 1633 Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1617, 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 1634 A closed version of one direction of 19.9 1637. (Contributed by NM, 5-Aug-1993.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
 
Theorem19.9t 1635 A closed version of 19.9 1637. (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 1636 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 1637 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 1638 One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1612. (Contributed by Jim Kingdon, 2-Jul-2018.)
(∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)
 
Theoremexnalim 1639 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 1640 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 1641 A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.)
(∀𝑥𝑦 ¬ 𝜑 → ¬ ∃𝑥𝑦𝜑)
 
Theoremnnal 1642 The double negation of a universal quantification implies the universal quantification of the double negation. (Contributed by BJ, 24-Nov-2023.)
(¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)
 
Theoremax6blem 1643 If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. This theorem doesn't use ax6b 1644 compared to hbnt 1646. (Contributed by GD, 27-Jan-2018.)
(𝜑 → ∀𝑥𝜑)       𝜑 → ∀𝑥 ¬ 𝜑)
 
Theoremax6b 1644 Quantified Negation. Axiom C5-2 of [Monk2] p. 113.

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

(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 
Theoremhbn1 1645 𝑥 is not free in ¬ ∀𝑥𝜑. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
 
Theoremhbnt 1646 Closed theorem version of bound-variable hypothesis builder hbn 1647. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
 
Theoremhbn 1647 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 1648 Deduction form of bound-variable hypothesis builder hbn 1647. (Contributed by NM, 3-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))
 
Theoremnfnt 1649 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 1650 Deduction associated with nfnt 1649. (Contributed by Mario Carneiro, 24-Sep-2016.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥 ¬ 𝜓)
 
Theoremnfn 1651 Inference associated with nfnt 1649. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥 ¬ 𝜑
 
Theoremnfdc 1652 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 1653 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 1654 A deduction version of one direction of 19.9 1637. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
(𝜓 → Ⅎ𝑥𝜑)       (𝜓 → (∃𝑥𝜑𝜑))
 
Theorem19.9hd 1655 A deduction version of one direction of 19.9 1637. This is an older variation of this theorem; new proofs should use 19.9d 1654. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝜓 → (𝜑 → ∀𝑥𝜑))       (𝜓 → (∃𝑥𝜑𝜑))
 
Theoremexcomim 1656 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
(∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑)
 
Theoremexcom 1657 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
(∃𝑥𝑦𝜑 ↔ ∃𝑦𝑥𝜑)
 
Theorem19.12 1658 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 1659 Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∃𝑥𝜓))
 
Theorem19.21-2 1660 Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)
𝑥𝜑    &   𝑦𝜑       (∀𝑥𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))
 
Theoremnf2 1661 An alternate definition of df-nf 1454, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
 
Theoremnf3 1662 An alternate definition of df-nf 1454. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑𝜑))
 
Theoremnf4dc 1663 Variable 𝑥 is effectively not free in 𝜑 iff 𝜑 is always true or always false, given a decidability condition. The reverse direction, nf4r 1664, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
(DECID𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)))
 
Theoremnf4r 1664 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 1663. (Contributed by Jim Kingdon, 21-Jul-2018.)
((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑)
 
Theorem19.36i 1665 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 1666 Closed form of 19.36i 1665. 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 1667 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 1668* Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
𝑥(𝜑𝜓)       (𝜑 → ∃𝑥𝜓)
 
Theorem19.38 1669 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
 
Theorem19.23t 1670 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 1671 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 1672 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 1673 One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if 𝜑 is decidable, as seen at 19.32dc 1672. (Contributed by Jim Kingdon, 28-Jul-2018.)
𝑥𝜑       ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
 
Theorem19.31r 1674 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 1675 Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
 
Theorem19.45 1676 Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
 
Theorem19.34 1677 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.41h 1678 Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1679 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 1679 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 1680 Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1681 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
 
Theorem19.42 1681 Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
 
Theoremexcom13 1682 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑧𝑦𝑥𝜑)
 
Theoremexrot3 1683 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑦𝑧𝑥𝜑)
 
Theoremexrot4 1684 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑧𝑤𝑥𝑦𝜑)
 
Theoremnexr 1685 Inference from 19.8a 1583. (Contributed by Jeff Hankins, 26-Jul-2009.)
¬ ∃𝑥𝜑        ¬ 𝜑
 
Theoremexan 1686 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 1687 Deduction form of bound-variable hypothesis builder hbex 1629. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → (∃𝑦𝜓 → ∀𝑥𝑦𝜓))
 
Theoremeeor 1688 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
 
1.3.8  Equality theorems without distinct variables
 
Theorema9e 1689 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 1440 through ax-14 2144 and ax-17 1519, all axioms other than ax-9 1524 are believed to be theorems of free logic, although the system without ax-9 1524 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
𝑥 𝑥 = 𝑦
 
Theorema9ev 1690* At least one individual exists. Weaker version of a9e 1689. (Contributed by NM, 3-Aug-2017.)
𝑥 𝑥 = 𝑦
 
Theoremax9o 1691 An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
 
Theoremspimfv 1692* Specialization, using implicit substitution. Version of spim 1731 with a disjoint variable condition. See spimv 1804 for another variant. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 31-May-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)
 
Theoremchvarfv 1693* Implicit substitution of 𝑦 for 𝑥 into a theorem. Version of chvar 1750 with a disjoint variable condition. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by BJ, 31-May-2019.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   𝜑       𝜓
 
Theoremequid 1694 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 1695 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 1696 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1763.) 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 1697 Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦𝑦 = 𝑥)
 
Theoremax6evr 1698* A commuted form of a9ev 1690. 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 1699 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
(𝑥 = 𝑦𝑦 = 𝑥)
 
Theoremequcomd 1700 Deduction form of equcom 1699, symmetry of equality. For the versions for classes, see eqcom 2172 and eqcomd 2176. (Contributed by BJ, 6-Oct-2019.)
(𝜑𝑥 = 𝑦)       (𝜑𝑦 = 𝑥)
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