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	r19.21bi 2401	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 2402	Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
⊢ ∀x ∈ A ∀y ∈ B φ    ⇒   ⊢ ((x ∈ A ∧ y ∈ B) → φ)
Theorem	rspec3 2403	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 2404	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 2405	Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
⊢ (x ∈ A → ¬ ψ)    ⇒   ⊢ ¬ ∃x ∈ A ψ
Theorem	nrexdv 2406*	Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
⊢ ((φ ∧ x ∈ A) → ¬ ψ)    ⇒   ⊢ (φ → ¬ ∃x ∈ A ψ)
Theorem	rexim 2407	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 2408	Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997.)
⊢ (x ∈ A → (φ → ψ))    ⇒   ⊢ (∃x ∈ A φ → ∃x ∈ A ψ)
Theorem	reximi2 2409	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 2410	Inference quantifying both antecedent and consequent. (Contributed by NM, 18-Oct-1996.)
⊢ (φ → ψ)    ⇒   ⊢ (∃x ∈ A φ → ∃x ∈ A ψ)
Theorem	reximdai 2411	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 2412*	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 2413*	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 2414*	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 2415*	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 2416*	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 2417	Closed theorem form of r19.23 2418. (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 2418	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 2419*	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 2420	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 2421*	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 2422*	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 2423*	Weaker version of rexlimiv 2421. (Contributed by FL, 19-Sep-2011.)
⊢ (φ → ψ)    ⇒   ⊢ (∃x ∈ A φ → ψ)
Theorem	rexlimd 2424	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 2425	Version of rexlimd 2424 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 2426*	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 2427*	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 2428*	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 2429*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2426. (Contributed by NM, 7-Jun-2015.)
⊢ ((φ ∧ x ∈ A ∧ ψ) → χ)    ⇒   ⊢ (φ → (∃x ∈ A ψ → χ))
Theorem	rexlimdvw 2430*	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 2431*	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 2432*	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 2433*	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 2434*	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 2435	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 2436	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 2437	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 2438	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 2439	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 2440	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 2441	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 2442*	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 2443*	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 2444	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 2445	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	r19.29af2 2446	A commonly used pattern based on r19.29 2444 (Contributed by Thierry Arnoux, 17-Dec-2017.)
⊢ Ⅎxφ    &   ⊢ Ⅎxχ    &   ⊢ (((φ ∧ x ∈ A) ∧ ψ) → χ)    &   ⊢ (φ → ∃x ∈ A ψ)    ⇒   ⊢ (φ → χ)
Theorem	r19.29af 2447*	A commonly used pattern based on r19.29 2444 (Contributed by Thierry Arnoux, 29-Nov-2017.)
⊢ Ⅎxφ    &   ⊢ (((φ ∧ x ∈ A) ∧ ψ) → χ)    &   ⊢ (φ → ∃x ∈ A ψ)    ⇒   ⊢ (φ → χ)
Theorem	r19.29a 2448*	A commonly used pattern based on r19.29 2444 (Contributed by Thierry Arnoux, 22-Nov-2017.)
⊢ (((φ ∧ x ∈ A) ∧ ψ) → χ)    &   ⊢ (φ → ∃x ∈ A ψ)    ⇒   ⊢ (φ → χ)
Theorem	r19.29d2r 2449	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 2450*	A commonly used pattern based on r19.29 2444, 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 2451	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	r19.32vr 2452*	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 2453. (Contributed by Jim Kingdon, 19-Aug-2018.)
⊢ ((φ ∨ ∀x ∈ A ψ) → ∀x ∈ A (φ ∨ ψ))
Theorem	r19.32vdc 2453*	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 2454	Restricted quantifier version of 19.35-1 1512. (Contributed by Jim Kingdon, 4-Jun-2018.)
⊢ (∃x ∈ A (φ → ψ) → (∀x ∈ A φ → ∃x ∈ A ψ))
Theorem	r19.36av 2455*	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 2456	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 2457*	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 2458	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 2459	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 2460*	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 2461*	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 2462	Restricted version of Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.) (Proof rewritten by Jim Kingdon, 5-Jun-2018.)
⊢ (∃x ∈ A (φ ∨ ψ) ↔ (∃x ∈ A φ ∨ ∃x ∈ A ψ))
Theorem	r19.44av 2463*	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 2464*	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 2465*	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 2466*	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 2467*	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 2468*	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 2469*	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 2470*	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	ralcom3 2471	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 2472*	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 2473*	Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
⊢ (∃x ∈ A ∃y ∈ B (φ ∧ ψ) ↔ (∃x ∈ A φ ∧ ∃y ∈ B ψ))
Theorem	3reeanv 2474*	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	nfreu1 2475	x is not free in ∃!x ∈ Aφ. (Contributed by NM, 19-Mar-1997.)
⊢ Ⅎx∃!x ∈ A φ
Theorem	nfrmo1 2476	x is not free in ∃*x ∈ Aφ. (Contributed by NM, 16-Jun-2017.)
⊢ Ⅎx∃*x ∈ A φ
Theorem	nfreudxy 2477*	Not-free deduction for restricted uniqueness. This is a version where x and y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
⊢ Ⅎyφ    &   ⊢ (φ → ℲxA)    &   ⊢ (φ → Ⅎxψ)    ⇒   ⊢ (φ → Ⅎx∃!y ∈ A ψ)
Theorem	nfreuxy 2478*	Not-free for restricted uniqueness. This is a version where x and y are distinct. (Contributed by Jim Kingdon, 6-Jun-2018.)
⊢ ℲxA    &   ⊢ Ⅎxφ    ⇒   ⊢ Ⅎx∃!y ∈ A φ
Theorem	rabid 2479	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 2480*	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 2481	Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2542. (Contributed by NM, 25-Nov-2013.)
⊢ (∀x ∈ A (ψ ↔ χ) ↔ {x ∈ A ∣ ψ} = {x ∈ A ∣ χ})
Theorem	rabswap 2482	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 2483	The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
⊢ Ⅎx{x ∈ A ∣ φ}
Theorem	nfrabxy 2484*	A variable not free in a wff remains so in a restricted class abstraction. (Contributed by Jim Kingdon, 19-Jul-2018.)
⊢ Ⅎxφ    &   ⊢ ℲxA    ⇒   ⊢ Ⅎx{y ∈ A ∣ φ}
Theorem	reubida 2485	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 2486*	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 2487*	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 2488	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 2489	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 2490	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 2491*	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 2492*	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))
Theorem	rmobiia 2493	Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
⊢ (x ∈ A → (φ ↔ ψ))    ⇒   ⊢ (∃*x ∈ A φ ↔ ∃*x ∈ A ψ)
Theorem	rmobii 2494	Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
⊢ (φ ↔ ψ)    ⇒   ⊢ (∃*x ∈ A φ ↔ ∃*x ∈ A ψ)
Theorem	raleqf 2495	Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ ℲxA    &   ⊢ ℲxB    ⇒   ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))
Theorem	rexeqf 2496	Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ ℲxA    &   ⊢ ℲxB    ⇒   ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ))
Theorem	reueq1f 2497	Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ ℲxA    &   ⊢ ℲxB    ⇒   ⊢ (A = B → (∃!x ∈ A φ ↔ ∃!x ∈ B φ))
Theorem	rmoeq1f 2498	Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ ℲxA    &   ⊢ ℲxB    ⇒   ⊢ (A = B → (∃*x ∈ A φ ↔ ∃*x ∈ B φ))
Theorem	raleq 2499*	Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))
Theorem	rexeq 2500*	Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ))