P. 28 of Theorem List - New Foundations Explorer

Theorem List for New Foundations Explorer - 2701-2800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	r19.21v 2701*	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 2702	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 2703*	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 2704*	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 2705*	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 2706*	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 2707*	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 2708*	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 2709*	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 2710*	Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)
⊢ ((x ∈ A ∧ y ∈ B) → φ)    ⇒   ⊢ ∀x ∈ A ∀y ∈ B φ
Theorem	rgen3 2711*	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 2712	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 2713	Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
⊢ ∀x ∈ A ∀y ∈ B φ    ⇒   ⊢ ((x ∈ A ∧ y ∈ B) → φ)
Theorem	rspec3 2714	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 2715	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 2716	Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
⊢ (x ∈ A → ¬ ψ)    ⇒   ⊢ ¬ ∃x ∈ A ψ
Theorem	nrexdv 2717*	Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
⊢ ((φ ∧ x ∈ A) → ¬ ψ)    ⇒   ⊢ (φ → ¬ ∃x ∈ A ψ)
Theorem	rexim 2718	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 2719	Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
⊢ (x ∈ A → (φ → ψ))    ⇒   ⊢ (∃x ∈ A φ → ∃x ∈ A ψ)
Theorem	reximi2 2720	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 2721	Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
⊢ (φ → ψ)    ⇒   ⊢ (∃x ∈ A φ → ∃x ∈ A ψ)
Theorem	reximdai 2722	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 2723*	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 2724*	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 2725*	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 2726*	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	r19.12 2727*	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 2728	Closed theorem form of r19.23 2729. (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 2729	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 2730*	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 2731	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 2732*	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 2733*	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 2734*	Weaker version of rexlimiv 2732. (Contributed by FL, 19-Sep-2011.)
⊢ (φ → ψ)    ⇒   ⊢ (∃x ∈ A φ → ψ)
Theorem	rexlimd 2735	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 2736	Version of rexlimd 2735 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 2737*	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 2738*	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 2739*	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 2740*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2737. (Contributed by NM, 7-Jun-2015.)
⊢ ((φ ∧ x ∈ A ∧ ψ) → χ)    ⇒   ⊢ (φ → (∃x ∈ A ψ → χ))
Theorem	rexlimdvw 2741*	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 2742*	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 2743*	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 2744*	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 2745*	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 2746	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	r19.26-2 2747	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 2748	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 2749	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 2750	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	ralbiim 2751	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 2752*	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 2753*	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 2754	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 2755	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	r19.30 2756	Theorem 19.30 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 25-Feb-2011.)
⊢ (∀x ∈ A (φ ∨ ψ) → (∀x ∈ A φ ∨ ∃x ∈ A ψ))
Theorem	r19.32v 2757*	Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 25-Nov-2003.)
⊢ (∀x ∈ A (φ ∨ ψ) ↔ (φ ∨ ∀x ∈ A ψ))
Theorem	r19.35 2758	Restricted quantifier version of Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 20-Sep-2003.)
⊢ (∃x ∈ A (φ → ψ) ↔ (∀x ∈ A φ → ∃x ∈ A ψ))
Theorem	r19.36av 2759*	One direction of a restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. The other direction doesn't hold when A is empty. (Contributed by NM, 22-Oct-2003.)
⊢ (∃x ∈ A (φ → ψ) → (∀x ∈ A φ → ψ))
Theorem	r19.37 2760	Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (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 2761*	Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 2-Apr-2004.)
⊢ (∃x ∈ A (φ → ψ) → (φ → ∃x ∈ A ψ))
Theorem	r19.40 2762	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 2763	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 2764*	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 2765*	Restricted version of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
⊢ (∃x ∈ A (φ ∧ ψ) ↔ (φ ∧ ∃x ∈ A ψ))
Theorem	r19.43 2766	Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ (∃x ∈ A (φ ∨ ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ))
Theorem	r19.44av 2767*	One direction of a restricted quantifier version of Theorem 19.44 of [Margaris] p. 90. The other direction doesn't hold when A is empty. (Contributed by NM, 2-Apr-2004.)
⊢ (∃x ∈ A (φ ∨ ψ) → (∃x ∈ A φ ∨ ψ))
Theorem	r19.45av 2768*	Restricted version of one direction of Theorem 19.45 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 2-Apr-2004.)
⊢ (∃x ∈ A (φ ∨ ψ) → (φ ∨ ∃x ∈ A ψ))
Theorem	ralcomf 2769*	Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
⊢ ℲyA    &   ⊢ ℲxB    ⇒   ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀y ∈ B ∀x ∈ A φ)
Theorem	rexcomf 2770*	Commutation of restricted quantifiers. (Contributed by Mario Carneiro, 14-Oct-2016.)
⊢ ℲyA    &   ⊢ ℲxB    ⇒   ⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃y ∈ B ∃x ∈ A φ)
Theorem	ralcom 2771*	Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀y ∈ B ∀x ∈ A φ)
Theorem	rexcom 2772*	Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
⊢ (∃x ∈ A ∃y ∈ B φ ↔ ∃y ∈ B ∃x ∈ A φ)
Theorem	rexcom13 2773*	Swap 1st and 3rd restricted existential quantifiers. (Contributed by NM, 8-Apr-2015.)
⊢ (∃x ∈ A ∃y ∈ B ∃z ∈ C φ ↔ ∃z ∈ C ∃y ∈ B ∃x ∈ A φ)
Theorem	rexrot4 2774*	Rotate existential restricted quantifiers twice. (Contributed by NM, 8-Apr-2015.)
⊢ (∃x ∈ A ∃y ∈ B ∃z ∈ C ∃w ∈ D φ ↔ ∃z ∈ C ∃w ∈ D ∃x ∈ A ∃y ∈ B φ)
Theorem	ralcom2 2775*	Commutation of restricted quantifiers. Note that x and y needn't be distinct (this makes the proof longer). (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)
⊢ (∀x ∈ A ∀y ∈ A φ → ∀y ∈ A ∀x ∈ A φ)
Theorem	ralcom3 2776	A commutative law for restricted quantifiers that swaps the domain of the restriction. (Contributed by NM, 22-Feb-2004.)
⊢ (∀x ∈ A (x ∈ B → φ) ↔ ∀x ∈ B (x ∈ A → φ))
Theorem	reean 2777*	Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ Ⅎyφ    &   ⊢ Ⅎxψ    ⇒   ⊢ (∃x ∈ A ∃y ∈ B (φ ∧ ψ) ↔ (∃x ∈ A φ ∧ ∃y ∈ B ψ))
Theorem	reeanv 2778*	Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
⊢ (∃x ∈ A ∃y ∈ B (φ ∧ ψ) ↔ (∃x ∈ A φ ∧ ∃y ∈ B ψ))
Theorem	3reeanv 2779*	Rearrange three existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)
⊢ (∃x ∈ A ∃y ∈ B ∃z ∈ C (φ ∧ ψ ∧ χ) ↔ (∃x ∈ A φ ∧ ∃y ∈ B ψ ∧ ∃z ∈ C χ))
Theorem	2ralor 2780*	Distribute quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.)
⊢ (∀x ∈ A ∀y ∈ B (φ ∨ ψ) ↔ (∀x ∈ A φ ∨ ∀y ∈ B ψ))
Theorem	nfreu1 2781	x is not free in ∃!x ∈ Aφ. (Contributed by NM, 19-Mar-1997.)
⊢ Ⅎx∃!x ∈ A φ
Theorem	nfrmo1 2782	x is not free in ∃*x ∈ Aφ. (Contributed by NM, 16-Jun-2017.)
⊢ Ⅎx∃*x ∈ A φ
Theorem	nfreud 2783	Deduction version of nfreu 2785. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.)
⊢ Ⅎyφ    &   ⊢ (φ → ℲxA)    &   ⊢ (φ → Ⅎxψ)    ⇒   ⊢ (φ → Ⅎx∃!y ∈ A ψ)
Theorem	nfrmod 2784	Deduction version of nfrmo 2786. (Contributed by NM, 17-Jun-2017.)
⊢ Ⅎyφ    &   ⊢ (φ → ℲxA)    &   ⊢ (φ → Ⅎxψ)    ⇒   ⊢ (φ → Ⅎx∃*y ∈ A ψ)
Theorem	nfreu 2785	Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.)
⊢ ℲxA    &   ⊢ Ⅎxφ    ⇒   ⊢ Ⅎx∃!y ∈ A φ
Theorem	nfrmo 2786	Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 16-Jun-2017.)
⊢ ℲxA    &   ⊢ Ⅎxφ    ⇒   ⊢ Ⅎx∃*y ∈ A φ
Theorem	rabid 2787	An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.)
⊢ (x ∈ {x ∈ A ∣ φ} ↔ (x ∈ A ∧ φ))
Theorem	rabid2 2788*	An "identity" law for restricted class abstraction. (Contributed by NM, 9-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
⊢ (A = {x ∈ A ∣ φ} ↔ ∀x ∈ A φ)
Theorem	rabbi 2789	Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2850. (Contributed by NM, 25-Nov-2013.)
⊢ (∀x ∈ A (ψ ↔ χ) ↔ {x ∈ A ∣ ψ} = {x ∈ A ∣ χ})
Theorem	rabswap 2790	Swap with a membership relation in a restricted class abstraction. (Contributed by NM, 4-Jul-2005.)
⊢ {x ∈ A ∣ x ∈ B} = {x ∈ B ∣ x ∈ A}
Theorem	nfrab1 2791	The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
⊢ Ⅎx{x ∈ A ∣ φ}
Theorem	nfrab 2792	A variable not free in a wff remains so in a restricted class abstraction. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 9-Oct-2016.)
⊢ Ⅎxφ    &   ⊢ ℲxA    ⇒   ⊢ Ⅎx{y ∈ A ∣ φ}
Theorem	reubida 2793	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by Mario Carneiro, 19-Nov-2016.)
⊢ Ⅎxφ    &   ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃!x ∈ A ψ ↔ ∃!x ∈ A χ))
Theorem	reubidva 2794*	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 13-Nov-2004.)
⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃!x ∈ A ψ ↔ ∃!x ∈ A χ))
Theorem	reubidv 2795*	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 17-Oct-1996.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃!x ∈ A ψ ↔ ∃!x ∈ A χ))
Theorem	reubiia 2796	Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 14-Nov-2004.)
⊢ (x ∈ A → (φ ↔ ψ))    ⇒   ⊢ (∃!x ∈ A φ ↔ ∃!x ∈ A ψ)
Theorem	reubii 2797	Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 22-Oct-1999.)
⊢ (φ ↔ ψ)    ⇒   ⊢ (∃!x ∈ A φ ↔ ∃!x ∈ A ψ)
Theorem	rmobida 2798	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
⊢ Ⅎxφ    &   ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃*x ∈ A ψ ↔ ∃*x ∈ A χ))
Theorem	rmobidva 2799*	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃*x ∈ A ψ ↔ ∃*x ∈ A χ))
Theorem	rmobidv 2800*	Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 16-Jun-2017.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃*x ∈ A ψ ↔ ∃*x ∈ A χ))