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

Theoremalrimd 1501 Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   𝑥𝜓    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremeximdh 1502 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Theoremeximd 1503 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Theoremnexd 1504 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)

Theoremexbidh 1505 Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremalbid 1506 Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Theoremexbid 1507 Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremexsimpl 1508 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 1509 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 1510 Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1536. (Contributed by Jim Kingdon, 2-Jun-2018.)
(∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑))

Theorem19.29 1511 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 1512 Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.29r2 1513 Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))

Theorem19.29x 1514 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 1515 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 1516 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 1517 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 1518 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 1519 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 1520 The antecedent provides a condition implying the converse of 19.33 1373. Compare Theorem 19.33 of [Margaris] p. 90. This variation of 19.33bdc 1521 is intuitionistically valid without a decidability condition. (Contributed by Mario Carneiro, 2-Feb-2015.)
((¬ ∃𝑥𝜑 ∨ ¬ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))

Theorem19.33bdc 1521 Converse of 19.33 1373 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 1520 (Contributed by Jim Kingdon, 23-Apr-2018.)
(DECID𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓))))

Theorem19.40 1522 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
(∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))

Theorem19.40-2 1523 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 1524 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑𝜓)))

Theoremexintr 1525 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))

Theoremalsyl 1526 Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)
((∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜒)) → ∀𝑥(𝜑𝜒))

Theoremhbex 1527 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 1528 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 1529 Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. (Contributed by O'Cat, 31-Mar-2008.)
(∀𝑥𝜑 → ∃𝑦𝜑)

Theoremi19.24 1530 Theorem 19.24 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1515, 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 1531 Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1515, 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 1532 A closed version of one direction of 19.9 1535. (Contributed by NM, 5-Aug-1993.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))

Theorem19.9t 1533 A closed version of 19.9 1535. (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 1534 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 1535 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 1536 One direction of theorem 19.6 of [Margaris] p. 89. The converse holds given a decidability condition, as seen at alexdc 1510. (Contributed by Jim Kingdon, 2-Jul-2018.)
(∀𝑥𝜑 → ¬ ∃𝑥 ¬ 𝜑)

Theoremexnalim 1537 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 1538 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 1539 A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.)
(∀𝑥𝑦 ¬ 𝜑 → ¬ ∃𝑥𝑦𝜑)

Theoremax6blem 1540 If 𝑥 is not free in 𝜑, it is not free in ¬ 𝜑. This theorem doesn't use ax6b 1541 compared to hbnt 1543. (Contributed by GD, 27-Jan-2018.)
(𝜑 → ∀𝑥𝜑)       𝜑 → ∀𝑥 ¬ 𝜑)

Theoremax6b 1541 Quantified Negation. Axiom C5-2 of [Monk2] p. 113.

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

(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremhbn1 1542 𝑥 is not free in ¬ ∀𝑥𝜑. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
(¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremhbnt 1543 Closed theorem version of bound-variable hypothesis builder hbn 1544. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
(∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Theoremhbn 1544 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 1545 Deduction form of bound-variable hypothesis builder hbn 1544. (Contributed by NM, 3-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → (¬ 𝜓 → ∀𝑥 ¬ 𝜓))

Theoremnfnt 1546 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 1547 Deduction associated with nfnt 1546. (Contributed by Mario Carneiro, 24-Sep-2016.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥 ¬ 𝜓)

Theoremnfn 1548 Inference associated with nfnt 1546. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑       𝑥 ¬ 𝜑

Theoremnfdc 1549 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 1550 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 1551 A deduction version of one direction of 19.9 1535. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
(𝜓 → Ⅎ𝑥𝜑)       (𝜓 → (∃𝑥𝜑𝜑))

Theorem19.9hd 1552 A deduction version of one direction of 19.9 1535. This is an older variation of this theorem; new proofs should use 19.9d 1551. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
(𝜓 → ∀𝑥𝜓)    &   (𝜓 → (𝜑 → ∀𝑥𝜑))       (𝜓 → (∃𝑥𝜑𝜑))

Theoremexcomim 1553 One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
(∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑)

Theoremexcom 1554 Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
(∃𝑥𝑦𝜑 ↔ ∃𝑦𝑥𝜑)

Theorem19.12 1555 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 1556 Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∃𝑥𝜓))

Theorem19.21-2 1557 Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)
𝑥𝜑    &   𝑦𝜑       (∀𝑥𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝑦𝜓))

Theoremnf2 1558 An alternative definition of df-nf 1350, which does not involve nested quantifiers on the same variable. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))

Theoremnf3 1559 An alternative definition of df-nf 1350. (Contributed by Mario Carneiro, 24-Sep-2016.)
(Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑𝜑))

Theoremnf4dc 1560 Variable 𝑥 is effectively not free in 𝜑 iff 𝜑 is always true or always false, given a decidability condition. The reverse direction, nf4r 1561, holds for all propositions. (Contributed by Jim Kingdon, 21-Jul-2018.)
(DECID𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)))

Theoremnf4r 1561 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 1560. (Contributed by Jim Kingdon, 21-Jul-2018.)
((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥𝜑)

Theorem19.36i 1562 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 1563 Closed form of 19.36i 1562. 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 1564 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 1565* Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
𝑥(𝜑𝜓)       (𝜑 → ∃𝑥𝜓)

Theorem19.38 1566 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))

Theorem19.23t 1567 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 1568 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 1569 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 1570 One direction of Theorem 19.32 of [Margaris] p. 90. The converse holds if 𝜑 is decidable, as seen at 19.32dc 1569. (Contributed by Jim Kingdon, 28-Jul-2018.)
𝑥𝜑       ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))

Theorem19.31r 1571 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 1572 Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜓       (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.45 1573 Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Theorem19.34 1574 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.41h 1575 Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1576 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 1576 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 1577 Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1578 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Theorem19.42 1578 Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
𝑥𝜑       (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Theoremexcom13 1579 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑧𝑦𝑥𝜑)

Theoremexrot3 1580 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧𝜑 ↔ ∃𝑦𝑧𝑥𝜑)

Theoremexrot4 1581 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦𝑧𝑤𝜑 ↔ ∃𝑧𝑤𝑥𝑦𝜑)

Theoremnexr 1582 Inference from 19.8a 1482. (Contributed by Jeff Hankins, 26-Jul-2009.)
¬ ∃𝑥𝜑        ¬ 𝜑

Theoremexan 1583 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 1584 Deduction form of bound-variable hypothesis builder hbex 1527. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑦𝜑)    &   (𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → (∃𝑦𝜓 → ∀𝑥𝑦𝜓))

Theoremeeor 1585 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))

1.3.8  Equality theorems without distinct variables

Theorema9e 1586 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 1336 through ax-14 1405 and ax-17 1419, all axioms other than ax-9 1424 are believed to be theorems of free logic, although the system without ax-9 1424 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
𝑥 𝑥 = 𝑦

Theorema9ev 1587* At least one individual exists. Weaker version of a9e 1586. (Contributed by NM, 3-Aug-2017.)
𝑥 𝑥 = 𝑦

Theoremax9o 1588 An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
(∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)

Theoremequid 1589 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 1590 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 1591 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1653.) 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 1592 Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremequcom 1593 Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremequcoms 1594 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 1595 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
(𝑥 = 𝑦 → (𝑦 = 𝑧𝑥 = 𝑧))

Theoremequtrr 1596 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 1597 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 1598 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremequequ2 1599 An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))

Theoremelequ1 1600 An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

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