P. 16 of Theorem List - Intuitionistic Logic Explorer

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

Type	Label	Description
Statement
Theorem	alrimd 1501	Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	eximdh 1502	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	eximd 1503	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	nexd 1504	Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ ∃𝑥𝜓)
Theorem	exbidh 1505	Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	albid 1506	Formula-building rule for universal quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	exbid 1507	Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	exsimpl 1508	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)
Theorem	exsimpr 1509	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)
Theorem	alexdc 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))
Theorem	19.29 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(𝜑 ∧ 𝜓))
Theorem	19.29r 1512	Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
Theorem	19.29r2 1513	Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.)
⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))
Theorem	19.29x 1514	Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
⊢ ((∃𝑥∀𝑦𝜑 ∧ ∀𝑥∃𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))
Theorem	19.35-1 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(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
Theorem	19.35i 1516	Inference from Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
Theorem	19.25 1517	Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) → (∃𝑦∀𝑥𝜑 → ∃𝑦∃𝑥𝜓))
Theorem	19.30dc 1518	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)))
Theorem	19.43 1519	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))
Theorem	19.33b2 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(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
Theorem	19.33bdc 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.)
⊢ (DECID ∃𝑥𝜑 → (¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓))))
Theorem	19.40 1522	Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
Theorem	19.40-2 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.)
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → (∃𝑥∃𝑦𝜑 ∧ ∃𝑥∃𝑦𝜓))
Theorem	exintrbi 1524	Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ 𝜓)))
Theorem	exintr 1525	Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))
Theorem	alsyl 1526	Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜒)) → ∀𝑥(𝜑 → 𝜒))
Theorem	hbex 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 → ∀𝑥∃𝑦𝜑)
Theorem	nfex 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.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃𝑦𝜑
Theorem	19.2 1529	Theorem 19.2 of [Margaris] p. 89, generalized to use two setvar variables. (Contributed by O'Cat, 31-Mar-2008.)
⊢ (∀𝑥𝜑 → ∃𝑦𝜑)
Theorem	i19.24 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(𝜑 → 𝜓))    ⇒   ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))
Theorem	i19.39 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(𝜑 → 𝜓))    ⇒   ⊢ ((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))
Theorem	19.9ht 1532	A closed version of one direction of 19.9 1535. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑))
Theorem	19.9t 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 → (∃𝑥𝜑 ↔ 𝜑))
Theorem	19.9h 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)    ⇒   ⊢ (∃𝑥𝜑 ↔ 𝜑)
Theorem	19.9 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.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 ↔ 𝜑)
Theorem	alexim 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)
Theorem	exnalim 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)
Theorem	exanaliim 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(𝜑 → 𝜓))
Theorem	alexnim 1539	A relationship between two quantifiers and negation. (Contributed by Jim Kingdon, 27-Aug-2018.)
⊢ (∀𝑥∃𝑦 ¬ 𝜑 → ¬ ∃𝑥∀𝑦𝜑)
Theorem	ax6blem 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)
Theorem	ax6b 1541	Quantified Negation. Axiom C5-2 of [Monk2] p. 113. (Contributed by GD, 27-Jan-2018.)
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Theorem	hbn1 1542	𝑥 is not free in ¬ ∀𝑥𝜑. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 18-Aug-2014.)
⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
Theorem	hbnt 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))
Theorem	hbn 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)
Theorem	hbnd 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))
Theorem	nfnt 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)
Theorem	nfnd 1547	Deduction associated with nfnt 1546. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
Theorem	nfn 1548	Inference associated with nfnt 1546. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥 ¬ 𝜑
Theorem	nfdc 1549	If 𝑥 is not free in 𝜑, it is not free in DECID 𝜑. (Contributed by Jim Kingdon, 11-Mar-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥DECID 𝜑
Theorem	modal-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)
Theorem	19.9d 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 → (∃𝑥𝜑 → 𝜑))
Theorem	19.9hd 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 → (∃𝑥𝜑 → 𝜑))
Theorem	excomim 1553	One direction of Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
Theorem	excom 1554	Theorem 19.11 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)
Theorem	19.12 1555	Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.)
⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑)
Theorem	19.19 1556	Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∃𝑥𝜓))
Theorem	19.21-2 1557	Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    ⇒   ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓))
Theorem	nf2 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))
Theorem	nf3 1559	An alternative definition of df-nf 1350. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(∃𝑥𝜑 → 𝜑))
Theorem	nf4dc 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.)
⊢ (DECID ∃𝑥𝜑 → (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)))
Theorem	nf4r 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)
Theorem	19.36i 1562	Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
⊢ Ⅎ𝑥𝜓    &   ⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	19.36-1 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.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	19.37-1 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.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))
Theorem	19.37aiv 1565*	Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	19.38 1566	Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
Theorem	19.23t 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(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓)))
Theorem	19.23 1568	Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))
Theorem	19.32dc 1569	Theorem 19.32 of [Margaris] p. 90, where 𝜑 is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (DECID 𝜑 → (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓)))
Theorem	19.32r 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.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ((𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓))
Theorem	19.31r 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.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ ((∀𝑥𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓))
Theorem	19.44 1572	Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ 𝜓))
Theorem	19.45 1573	Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
Theorem	19.34 1574	Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ ((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑 ∨ 𝜓))
Theorem	19.41h 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(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
Theorem	19.41 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.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ 𝜓))
Theorem	19.42h 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(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
Theorem	19.42 1578	Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
Theorem	excom13 1579	Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑧∃𝑦∃𝑥𝜑)
Theorem	exrot3 1580	Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑦∃𝑧∃𝑥𝜑)
Theorem	exrot4 1581	Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
⊢ (∃𝑥∃𝑦∃𝑧∃𝑤𝜑 ↔ ∃𝑧∃𝑤∃𝑥∃𝑦𝜑)
Theorem	nexr 1582	Inference from 19.8a 1482. (Contributed by Jeff Hankins, 26-Jul-2009.)
⊢ ¬ ∃𝑥𝜑    ⇒   ⊢ ¬ 𝜑
Theorem	exan 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.)
⊢ (∃𝑥𝜑 ∧ 𝜓)    ⇒   ⊢ ∃𝑥(𝜑 ∧ 𝜓)
Theorem	hbexd 1584	Deduction form of bound-variable hypothesis builder hbex 1527. (Contributed by NM, 2-Jan-2002.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → (∃𝑦𝜓 → ∀𝑥∃𝑦𝜓))
Theorem	eeor 1585	Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥∃𝑦(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
1.3.8  Equality theorems without distinct variables
Theorem	a9e 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
Theorem	a9ev 1587*	At least one individual exists. Weaker version of a9e 1586. (Contributed by NM, 3-Aug-2017.)
⊢ ∃𝑥 𝑥 = 𝑦
Theorem	ax9o 1588	An implication related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 3-Feb-2015.)
⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑)
Theorem	equid 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
Theorem	nfequid 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.)
⊢ Ⅎ𝑦 𝑥 = 𝑥
Theorem	stdpc6 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
Theorem	equcomi 1592	Commutative law for equality. Lemma 7 of [Tarski] p. 69. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
Theorem	equcom 1593	Commutative law for equality. (Contributed by NM, 20-Aug-1993.)
⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
Theorem	equcoms 1594	An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → 𝜑)    ⇒   ⊢ (𝑦 = 𝑥 → 𝜑)
Theorem	equtr 1595	A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧))
Theorem	equtrr 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 = 𝑥 → 𝑧 = 𝑦))
Theorem	equtr2 1597	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)
Theorem	equequ1 1598	An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
Theorem	equequ2 1599	An equivalence law for equality. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑦))
Theorem	elequ1 1600	An identity law for the non-logical predicate. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))