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	alimd 1501	Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	alrimi 1502	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	nfd 1503	Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	nfdh 1504	Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	nfrimi 1505	Moving an antecedent outside Ⅎ. (Contributed by Jim Kingdon, 23-Mar-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥(𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
1.3.3  Axiom ax-17 - first use of the $d distinct variable statement
Axiom	ax-17 1506*	Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)
Theorem	a17d 1507*	ax-17 1506 with antecedent. (Contributed by NM, 1-Mar-2013.)
⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
Theorem	nfv 1508*	If 𝑥 is not present in 𝜑, then 𝑥 is not free in 𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑
Theorem	nfvd 1509*	nfv 1508 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1564. (Contributed by Mario Carneiro, 6-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)
1.3.4  Introduce Axiom of Existence
Axiom	ax-i9 1510	Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. One thing this axiom tells us is that at least one thing exists (although ax-4 1487 and possibly others also tell us that, i.e. they are not valid in the empty domain of a "free logic"). In this form (not requiring that 𝑥 and 𝑦 be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) Another name for this theorem is a9e 1674, which has additional remarks. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ ∃𝑥 𝑥 = 𝑦
Theorem	ax-9 1511	Derive ax-9 1511 from ax-i9 1510, the modified version for intuitionistic logic. Although ax-9 1511 does hold intuistionistically, in intuitionistic logic it is weaker than ax-i9 1510. (Contributed by NM, 3-Feb-2015.)
⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦
Theorem	equidqe 1512	equid 1677 with some quantification and negation without using ax-4 1487 or ax-17 1506. (Contributed by NM, 13-Jan-2011.) (Proof shortened by Wolf Lammen, 27-Feb-2014.)
⊢ ¬ ∀𝑦 ¬ 𝑥 = 𝑥
Theorem	ax4sp1 1513	A special case of ax-4 1487 without using ax-4 1487 or ax-17 1506. (Contributed by NM, 13-Jan-2011.)
⊢ (∀𝑦 ¬ 𝑥 = 𝑥 → ¬ 𝑥 = 𝑥)
1.3.5  Additional intuitionistic axioms
Axiom	ax-ial 1514	𝑥 is not free in ∀𝑥𝜑. One of the axioms of predicate logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Axiom	ax-i5r 1515	Axiom of quantifier collection. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓))
1.3.6  Predicate calculus including ax-4, without distinct variables
Theorem	spi 1516	Inference reversing generalization (specialization). (Contributed by NM, 5-Aug-1993.)
⊢ ∀𝑥𝜑    ⇒   ⊢ 𝜑
Theorem	sps 1517	Generalization of antecedent. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → 𝜓)
Theorem	spsd 1518	Deduction generalizing antecedent. (Contributed by NM, 17-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))
Theorem	nfbidf 1519	An equality theorem for effectively not free. (Contributed by Mario Carneiro, 4-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))
Theorem	hba1 1520	𝑥 is not free in ∀𝑥𝜑. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)
Theorem	nfa1 1521	𝑥 is not free in ∀𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥∀𝑥𝜑
Theorem	a5i 1522	Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)
Theorem	nfnf1 1523	𝑥 is not free in Ⅎ𝑥𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥Ⅎ𝑥𝜑
Theorem	hbim 1524	If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 → 𝜓). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))
Theorem	hbor 1525	If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∨ 𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ ((𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓))
Theorem	hban 1526	If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))
Theorem	hbbi 1527	If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ↔ 𝜓). (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ ((𝜑 ↔ 𝜓) → ∀𝑥(𝜑 ↔ 𝜓))
Theorem	hb3or 1528	If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∨ 𝜓 ∨ 𝜒). (Contributed by NM, 14-Sep-2003.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) → ∀𝑥(𝜑 ∨ 𝜓 ∨ 𝜒))
Theorem	hb3an 1529	If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (Contributed by NM, 14-Sep-2003.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝜒 → ∀𝑥𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
Theorem	hba2 1530	Lemma 24 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦∀𝑥𝜑)
Theorem	hbia1 1531	Lemma 23 of [Monk2] p. 114. (Contributed by NM, 29-May-2008.)
⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	19.3h 1532	A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-May-2007.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∀𝑥𝜑 ↔ 𝜑)
Theorem	19.3 1533	A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥𝜑 ↔ 𝜑)
Theorem	19.16 1534	Theorem 19.16 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∀𝑥𝜓))
Theorem	19.17 1535	Theorem 19.17 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ 𝜓))
Theorem	19.21h 1536	Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." New proofs should use 19.21 1562 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Theorem	19.21bi 1537	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	19.21bbi 1538	Inference removing double quantifier. (Contributed by NM, 20-Apr-1994.)
⊢ (𝜑 → ∀𝑥∀𝑦𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	19.27h 1539	Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))
Theorem	19.27 1540	Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))
Theorem	19.28h 1541	Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Theorem	19.28 1542	Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
Theorem	nfan1 1543	A closed form of nfan 1544. (Contributed by Mario Carneiro, 3-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)
Theorem	nfan 1544	If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∧ 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)
Theorem	nf3an 1545	If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝜒    ⇒   ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓 ∧ 𝜒)
Theorem	nford 1546	If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓 ∨ 𝜒). (Contributed by Jim Kingdon, 29-Oct-2019.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∨ 𝜒))
Theorem	nfand 1547	If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓 ∧ 𝜒). (Contributed by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒))
Theorem	nf3and 1548	Deduction form of bound-variable hypothesis builder nf3an 1545. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → Ⅎ𝑥𝜃)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒 ∧ 𝜃))
Theorem	hbim1 1549	A closed form of hbim 1524. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    ⇒   ⊢ ((𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))
Theorem	nfim1 1550	A closed form of nfim 1551. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ Ⅎ𝑥(𝜑 → 𝜓)
Theorem	nfim 1551	If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 → 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥(𝜑 → 𝜓)
Theorem	hbimd 1552	Deduction form of bound-variable hypothesis builder hbim 1524. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒)))
Theorem	nfor 1553	If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑 ∨ 𝜓). (Contributed by Jim Kingdon, 11-Mar-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓)
Theorem	hbbid 1554	Deduction form of bound-variable hypothesis builder hbbi 1527. (Contributed by NM, 1-Jan-2002.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    ⇒   ⊢ (𝜑 → ((𝜓 ↔ 𝜒) → ∀𝑥(𝜓 ↔ 𝜒)))
Theorem	nfal 1555	If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∀𝑦𝜑
Theorem	nfnf 1556	If 𝑥 is not free in 𝜑, it is not free in Ⅎ𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥Ⅎ𝑦𝜑
Theorem	nfalt 1557	Closed form of nfal 1555. (Contributed by Jim Kingdon, 11-May-2018.)
⊢ (∀𝑦Ⅎ𝑥𝜑 → Ⅎ𝑥∀𝑦𝜑)
Theorem	nfa2 1558	Lemma 24 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥∀𝑦∀𝑥𝜑
Theorem	nfia1 1559	Lemma 23 of [Monk2] p. 114. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥(∀𝑥𝜑 → ∀𝑥𝜓)
Theorem	19.21ht 1560	Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (New usage is discouraged.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Theorem	19.21t 1561	Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
Theorem	19.21 1562	Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Theorem	stdpc5 1563	An axiom scheme of standard predicate calculus that emulates Axiom 5 of [Mendelson] p. 69. The hypothesis Ⅎ𝑥𝜑 can be thought of as emulating "𝑥 is not free in 𝜑." With this definition, the meaning of "not free" is less restrictive than the usual textbook definition; for example 𝑥 would not (for us) be free in 𝑥 = 𝑥 by nfequid 1678. This theorem scheme can be proved as a metatheorem of Mendelson's axiom system, even though it is slightly stronger than his Axiom 5. (Contributed by NM, 22-Sep-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))
Theorem	nfimd 1564	If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 → 𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒))
Theorem	aaanh 1565	Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
⊢ (𝜑 → ∀𝑦𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
Theorem	aaan 1566	Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
Theorem	nfbid 1567	If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 ↔ 𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝜒))
Theorem	nfbi 1568	If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 ↔ 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥(𝜑 ↔ 𝜓)
1.3.7  The existential quantifier
Theorem	19.8a 1569	If a wff is true, then it is true for at least one instance. Special case of Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∃𝑥𝜑)
Theorem	19.8ad 1570	If a wff is true, it is true for at least one instance. Deduction form of 19.8a 1569. (Contributed by DAW, 13-Feb-2017.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥𝜓)
Theorem	19.23bi 1571	Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (∃𝑥𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	exlimih 1572	Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	exlimi 1573	Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	exlimd2 1574	Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1575 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
Theorem	exlimdh 1575	Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜒 → ∀𝑥𝜒)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
Theorem	exlimd 1576	Deduction from Theorem 19.9 of [Margaris] p. 89. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 18-Jun-2018.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒))
Theorem	exlimiv 1577*	Inference from Theorem 19.23 of [Margaris] p. 90. This inference, along with our many variants is used to implement a metatheorem called "Rule C" that is given in many logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or Rule C in Hirst and Hirst's A Primer for Logic and Proof p. 59 (PDF p. 65) at http://www.mathsci.appstate.edu/~jlh/primer/hirst.pdf. In informal proofs, the statement "Let C be an element such that..." almost always means an implicit application of Rule C. In essence, Rule C states that if we can prove that some element 𝑥 exists satisfying a wff, i.e. ∃𝑥𝜑(𝑥) where 𝜑(𝑥) has 𝑥 free, then we can use 𝜑( C ) as a hypothesis for the proof where C is a new (ficticious) constant not appearing previously in the proof, nor in any axioms used, nor in the theorem to be proved. The purpose of Rule C is to get rid of the existential quantifier. We cannot do this in Metamath directly. Instead, we use the original 𝜑 (containing 𝑥) as an antecedent for the main part of the proof. We eventually arrive at (𝜑 → 𝜓) where 𝜓 is the theorem to be proved and does not contain 𝑥. Then we apply exlimiv 1577 to arrive at (∃𝑥𝜑 → 𝜓). Finally, we separately prove ∃𝑥𝜑 and detach it with modus ponens ax-mp 5 to arrive at the final theorem 𝜓. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 25-Jul-2012.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → 𝜓)
Theorem	exim 1578	Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
Theorem	eximi 1579	Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑥𝜓)
Theorem	2eximi 1580	Inference adding 2 existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓)
Theorem	eximii 1581	Inference associated with eximi 1579. (Contributed by BJ, 3-Feb-2018.)
⊢ ∃𝑥𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ ∃𝑥𝜓
Theorem	alinexa 1582	A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓))
Theorem	exbi 1583	Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
Theorem	exbii 1584	Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑥𝜓)
Theorem	2exbii 1585	Inference adding 2 existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓)
Theorem	3exbii 1586	Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓)
Theorem	exancom 1587	Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
Theorem	alrimdd 1588	Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	alrimd 1589	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 1590	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 1591	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 1592	Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ ∃𝑥𝜓)
Theorem	exbidh 1593	Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	albid 1594	Formula-building rule for universal quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	exbid 1595	Formula-building rule for existential quantifier (deduction form). (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	exsimpl 1596	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 1597	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 1598	Theorem 19.6 of [Margaris] p. 89, given a decidability condition. The forward direction holds for all propositions, as seen at alexim 1624. (Contributed by Jim Kingdon, 2-Jun-2018.)
⊢ (∀𝑥DECID 𝜑 → (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑))
Theorem	19.29 1599	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 1600	Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))