P. 27 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2601-2700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ralim 2601	Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓))
Theorem	ralimi2 2602	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 2603	Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	ralimiaa 2604	Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
Theorem	ralimi 2605	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 2606	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 2607	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 2608	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 2609*	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 2610*	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 2611*	Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1506). (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 2612*	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 2613	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 2614*	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 2615*	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 2616*	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 2617	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 2618	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 2619*	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 2620	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 2621*	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 2622*	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 2623*	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 2624*	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 2625*	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 2626*	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 2627*	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 2628*	Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)    ⇒   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑
Theorem	rgen3 2629*	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 2630	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 2631	Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑    ⇒   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)
Theorem	rspec3 2632	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 2633	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 2634	Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
⊢ (𝑥 ∈ 𝐴 → ¬ 𝜓)    ⇒   ⊢ ¬ ∃𝑥 ∈ 𝐴 𝜓
Theorem	nrexdv 2635*	Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝜓)
Theorem	rexim 2636	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 2637	Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	reximi2 2638	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 2639	Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
Theorem	reximdai 2640	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 2641*	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 2642*	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 2643*	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 2644*	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 2645*	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 2646*	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 2647*	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	rexanaliim 2648	A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) (Revised by Jim Kingdon, 18-Jan-2026.)
⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ ¬ 𝜓) → ¬ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓))
Theorem	r19.12 2649*	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 2650	Closed theorem form of r19.23 2651. (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 2651	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 2652*	Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))
Theorem	rexlimi 2653	Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)
Theorem	rexlimiv 2654*	Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)
⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)
Theorem	rexlimiva 2655*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)
Theorem	rexlimivw 2656*	Weaker version of rexlimiv 2654. (Contributed by FL, 19-Sep-2011.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)
Theorem	rexlimd 2657	Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
Theorem	rexlimd2 2658	Version of rexlimd 2657 with deduction version of second hypothesis. (Contributed by NM, 21-Jul-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
Theorem	rexlimdv 2659*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 14-Nov-2002.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
Theorem	rexlimdva 2660*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 20-Jan-2007.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
Theorem	rexlimdvaa 2661*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
Theorem	rexlimdv3a 2662*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2659. (Contributed by NM, 7-Jun-2015.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
Theorem	rexlimdva2 2663*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
Theorem	rexlimdvw 2664*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Jun-2014.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
Theorem	rexlimddv 2665*	Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	rexlimivv 2666*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝜓)
Theorem	rexlimdvv 2667*	Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒))
Theorem	rexlimdvva 2668*	Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒))
Theorem	r19.26 2669	Theorem 19.26 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 28-Jan-1997.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	r19.27v 2670*	Restricted quantitifer version of one direction of 19.27 1610. (The other direction holds when 𝐴 is inhabited, see r19.27mv 3605.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
Theorem	r19.28v 2671*	Restricted quantifier version of one direction of 19.28 1612. (The other direction holds when 𝐴 is inhabited, see r19.28mv 3601.) (Contributed by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
Theorem	r19.26-2 2672	Theorem 19.26 of [Margaris] p. 90 with 2 restricted quantifiers. (Contributed by NM, 10-Aug-2004.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓))
Theorem	r19.26-3 2673	Theorem 19.26 of [Margaris] p. 90 with 3 restricted quantifiers. (Contributed by FL, 22-Nov-2010.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 𝜒))
Theorem	r19.26m 2674	Theorem 19.26 of [Margaris] p. 90 with mixed quantifiers. (Contributed by NM, 22-Feb-2004.)
⊢ (∀𝑥((𝑥 ∈ 𝐴 → 𝜑) ∧ (𝑥 ∈ 𝐵 → 𝜓)) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜓))
Theorem	ralbi 2675	Distribute a restricted universal quantifier over a biconditional. Theorem 19.15 of [Margaris] p. 90 with restricted quantification. (Contributed by NM, 6-Oct-2003.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	rexbi 2676	Distribute a restricted existential quantifier over a biconditional. Theorem 19.18 of [Margaris] p. 90 with restricted quantification. (Contributed by Jim Kingdon, 21-Jan-2019.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜓))
Theorem	ralbiim 2677	Split a biconditional and distribute quantifier. (Contributed by NM, 3-Jun-2012.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝜓) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)))
Theorem	r19.27av 2678*	Restricted version of one direction of Theorem 19.27 of [Margaris] p. 90. (The other direction doesn't hold when 𝐴 is empty.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
Theorem	r19.28av 2679*	Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when 𝐴 is empty.) (Contributed by NM, 2-Apr-2004.)
⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
Theorem	r19.29 2680	Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
Theorem	r19.29r 2681	Variation of Theorem 19.29 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 31-Aug-1999.)
⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
Theorem	ralnex2 2682	Relationship between two restricted universal and existential quantifiers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 18-May-2023.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)
Theorem	r19.29af2 2683	A commonly used pattern based on r19.29 2680. (Contributed by Thierry Arnoux, 17-Dec-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝜒    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	r19.29af 2684*	A commonly used pattern based on r19.29 2680. (Contributed by Thierry Arnoux, 29-Nov-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	r19.29an 2685*	A commonly used pattern based on r19.29 2680. (Contributed by Thierry Arnoux, 29-Dec-2019.)
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → 𝜒)
Theorem	r19.29a 2686*	A commonly used pattern based on r19.29 2680. (Contributed by Thierry Arnoux, 22-Nov-2017.)
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	r19.29d2r 2687	Theorem 19.29 of [Margaris] p. 90 with two restricted quantifiers, deduction version. (Contributed by Thierry Arnoux, 30-Jan-2017.)
⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜓 ∧ 𝜒))
Theorem	r19.29vva 2688*	A commonly used pattern based on r19.29 2680, version with two restricted quantifiers. (Contributed by Thierry Arnoux, 26-Nov-2017.)
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) → 𝜒)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	r19.32r 2689	One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence. (Contributed by Jim Kingdon, 19-Aug-2018.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓))
Theorem	r19.30dc 2690	Restricted quantifier version of 19.30dc 1676. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by Wolf Lammen, 18-Jun-2023.)
⊢ ((∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ∧ DECID ∃𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐴 𝜓))
Theorem	r19.32vr 2691*	One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence, as seen at r19.32vdc 2692. (Contributed by Jim Kingdon, 19-Aug-2018.)
⊢ ((𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓))
Theorem	r19.32vdc 2692*	Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, where 𝜑 is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.)
⊢ (DECID 𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥 ∈ 𝐴 𝜓)))
Theorem	r19.35-1 2693	Restricted quantifier version of 19.35-1 1673. (Contributed by Jim Kingdon, 4-Jun-2018.)
⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
Theorem	r19.36av 2694*	One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. In classical logic, the converse would hold if 𝐴 has at least one element, but in intuitionistic logic, that is not a sufficient condition. (Contributed by NM, 22-Oct-2003.)
⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → 𝜓))
Theorem	r19.37 2695	Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. In classical logic the converse would hold if 𝐴 has at least one element, but that is not sufficient in intuitionistic logic. (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
Theorem	r19.37av 2696*	Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∃𝑥 ∈ 𝐴 𝜓))
Theorem	r19.40 2697	Restricted quantifier version of Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 2-Apr-2004.)
⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) → (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓))
Theorem	r19.41 2698	Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)
⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))
Theorem	r19.41v 2699*	Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 17-Dec-2003.)
⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ 𝜓))
Theorem	r19.42v 2700*	Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓))