P. 25 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2401-2500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rexbii 2401	Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario Carneiro, 17-Oct-2016.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓)
Theorem	2ralbii 2402	Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)
Theorem	2rexbii 2403	Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)
Theorem	ralbii2 2404	Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)
⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐵 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)
Theorem	rexbii2 2405	Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜓)
Theorem	raleqbii 2406	Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)
Theorem	rexeqbii 2407	Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒)
Theorem	ralbiia 2408	Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)
Theorem	rexbiia 2409	Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓)
Theorem	2rexbiia 2410*	Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)
Theorem	r2alf 2411*	Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑦𝐴    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))
Theorem	r2exf 2412*	Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑦𝐴    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))
Theorem	r2al 2413*	Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))
Theorem	r2ex 2414*	Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))
Theorem	2ralbida 2415*	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 24-Feb-2004.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))
Theorem	2ralbidva 2416*	Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 4-Mar-1997.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))
Theorem	2rexbidva 2417*	Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 15-Dec-2004.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒))
Theorem	2ralbidv 2418*	Formula-building rule for restricted universal quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))
Theorem	2rexbidv 2419*	Formula-building rule for restricted existential quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒))
Theorem	rexralbidv 2420*	Formula-building rule for restricted quantifiers (deduction form). (Contributed by NM, 28-Jan-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))
Theorem	ralinexa 2421	A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
Theorem	risset 2422*	Two ways to say "𝐴 belongs to 𝐵." (Contributed by NM, 22-Nov-1994.)
⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
Theorem	hbral 2423	Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    &   ⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ (∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥∀𝑦 ∈ 𝐴 𝜑)
Theorem	hbra1 2424	𝑥 is not free in ∀𝑥 ∈ 𝐴𝜑. (Contributed by NM, 18-Oct-1996.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥∀𝑥 ∈ 𝐴 𝜑)
Theorem	nfra1 2425	𝑥 is not free in ∀𝑥 ∈ 𝐴𝜑. (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑
Theorem	nfraldxy 2426*	Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfraldya 2428 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 29-May-2018.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)
Theorem	nfrexdxy 2427*	Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexdya 2429 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)
Theorem	nfraldya 2428*	Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfraldxy 2426 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)
Theorem	nfrexdya 2429*	Not-free for restricted existential quantification where 𝑦 and 𝐴 are distinct. See nfrexdxy 2427 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)
Theorem	nfralxy 2430*	Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfralya 2432 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑
Theorem	nfrexxy 2431*	Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexya 2433 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑
Theorem	nfralya 2432*	Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfralxy 2430 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑
Theorem	nfrexya 2433*	Not-free for restricted existential quantification where 𝑦 and 𝐴 are distinct. See nfrexxy 2431 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑
Theorem	nfra2xy 2434*	Not-free given two restricted quantifiers. (Contributed by Jim Kingdon, 20-Aug-2018.)
⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
Theorem	nfre1 2435	𝑥 is not free in ∃𝑥 ∈ 𝐴𝜑. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜑
Theorem	r3al 2436*	Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜑))
Theorem	alral 2437	Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	rexex 2438	Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥𝜑)
Theorem	rsp 2439	Restricted specialization. (Contributed by NM, 17-Oct-1996.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))
Theorem	rspe 2440	Restricted specialization. (Contributed by NM, 12-Oct-1999.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 ∈ 𝐴 𝜑)
Theorem	rsp2 2441	Restricted specialization. (Contributed by NM, 11-Feb-1997.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))
Theorem	rsp2e 2442	Restricted specialization. (Contributed by FL, 4-Jun-2012.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
Theorem	rspec 2443	Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
⊢ ∀𝑥 ∈ 𝐴 𝜑    ⇒   ⊢ (𝑥 ∈ 𝐴 → 𝜑)
Theorem	rgen 2444	Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
⊢ (𝑥 ∈ 𝐴 → 𝜑)    ⇒   ⊢ ∀𝑥 ∈ 𝐴 𝜑
Theorem	rgen2a 2445*	Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct (and illustrates the use of dvelimor 1954). (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑)    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑
Theorem	rgenw 2446	Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
⊢ 𝜑    ⇒   ⊢ ∀𝑥 ∈ 𝐴 𝜑
Theorem	rgen2w 2447	Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct. (Contributed by NM, 18-Jun-2014.)
⊢ 𝜑    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
Theorem	mprg 2448	Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → 𝜓)    &   ⊢ (𝑥 ∈ 𝐴 → 𝜑)    ⇒   ⊢ 𝜓
Theorem	mprgbir 2449	Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)
⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)    &   ⊢ (𝑥 ∈ 𝐴 → 𝜓)    ⇒   ⊢ 𝜑
Theorem	ralim 2450	Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
Theorem	ralimi2 2451	Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐵 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐵 𝜓)
Theorem	ralimia 2452	Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	ralimiaa 2453	Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	ralimi 2454	Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	2ralimi 2455	Inference quantifying both antecedent and consequent two times, with strong hypothesis. (Contributed by AV, 3-Dec-2021.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)
Theorem	ral2imi 2456	Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
Theorem	ralimdaa 2457	Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
Theorem	ralimdva 2458*	Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
Theorem	ralimdv 2459*	Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
Theorem	ralimdvva 2460*	Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1401). (Contributed by AV, 27-Nov-2019.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))
Theorem	ralimdv2 2461*	Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)
⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) → (𝑥 ∈ 𝐵 → 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐵 𝜒))
Theorem	ralrimi 2462	Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	ralrimiv 2463*	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	ralrimiva 2464*	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Jan-2006.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	ralrimivw 2465*	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	r19.21t 2466	Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)
⊢ (Ⅎ𝑥𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)))
Theorem	r19.21 2467	Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 30-Mar-2011.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
Theorem	r19.21v 2468*	Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
Theorem	ralrimd 2469	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜑 → (𝜓 → (𝑥 ∈ 𝐴 → 𝜒)))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
Theorem	ralrimdv 2470*	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998.)
⊢ (𝜑 → (𝜓 → (𝑥 ∈ 𝐴 → 𝜒)))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
Theorem	ralrimdva 2471*	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒))
Theorem	ralrimivv 2472*	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜓))    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)
Theorem	ralrimivva 2473*	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)
Theorem	ralrimivvva 2474*	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜓)
Theorem	ralrimdvv 2475*	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)
⊢ (𝜑 → (𝜓 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜒)))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))
Theorem	ralrimdvva 2476*	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜒))
Theorem	rgen2 2477*	Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
Theorem	rgen3 2478*	Generalization rule for restricted quantification. (Contributed by NM, 12-Jan-2008.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜑)    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑
Theorem	r19.21bi 2479	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
Theorem	rspec2 2480	Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑    ⇒   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)
Theorem	rspec3 2481	Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑    ⇒   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜑)
Theorem	r19.21be 2482	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)
Theorem	nrex 2483	Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
⊢ (𝑥 ∈ 𝐴 → ¬ 𝜓)    ⇒   ⊢ ¬ ∃𝑥 ∈ 𝐴 𝜓
Theorem	nrexdv 2484*	Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝜓)
Theorem	rexim 2485	Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
Theorem	reximia 2486	Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	reximi2 2487	Inference quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 8-Nov-2004.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
Theorem	reximi 2488	Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	reximdai 2489	Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))
Theorem	reximdv2 2490*	Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 17-Sep-2003.)
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) → (𝑥 ∈ 𝐵 ∧ 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐵 𝜒))
Theorem	reximdvai 2491*	Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))
Theorem	reximdv 2492*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version with strong hypothesis.) (Contributed by NM, 24-Jun-1998.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))
Theorem	reximdva 2493*	Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))
Theorem	reximddv 2494*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 7-Dec-2016.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)
Theorem	reximssdv 2495*	Derivation of a restricted existential quantification over a subset (the second hypothesis implies 𝐴 ⊆ 𝐵), deduction form. (Contributed by AV, 21-Aug-2022.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → 𝑥 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)
Theorem	reximddv2 2496*	Double deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 15-Dec-2019.)
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)
Theorem	r19.12 2497*	Theorem 19.12 of [Margaris] p. 89 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑)
Theorem	r19.23t 2498	Closed theorem form of r19.23 2499. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
⊢ (Ⅎ𝑥𝜓 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)))
Theorem	r19.23 2499	Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))
Theorem	r19.23v 2500*	Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))