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	r19.23t 2401	Closed theorem form of r19.23 2402. (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 2402	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 2403*	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 2404	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 2405*	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 2406*	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 2407*	Weaker version of rexlimiv 2405. (Contributed by FL, 19-Sep-2011.)
⊢ (φ → ψ)    ⇒   ⊢ (∃x ∈ A φ → ψ)
Theorem	rexlimd 2408	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 2409	Version of rexlimd 2408 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 2410*	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 2411*	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 2412*	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 2413*	Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). Frequently-used variant of rexlimdv 2410. (Contributed by NM, 7-Jun-2015.)
⊢ ((φ ∧ x ∈ A ∧ ψ) → χ)    ⇒   ⊢ (φ → (∃x ∈ A ψ → χ))
Theorem	rexlimdvw 2414*	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 2415*	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 2416*	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 2417*	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 2418*	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 2419	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 2420	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 2421	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 2422	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 2423	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 2424	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 2425	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 2426*	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 2427*	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 2428	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 2429	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 2430	A commonly used pattern based on r19.29 2428 (Contributed by Thierry Arnoux, 17-Dec-2017.)
⊢ Ⅎxφ    &   ⊢ Ⅎxχ    &   ⊢ (((φ ∧ x ∈ A) ∧ ψ) → χ)    &   ⊢ (φ → ∃x ∈ A ψ)    ⇒   ⊢ (φ → χ)
Theorem	r19.29af 2431*	A commonly used pattern based on r19.29 2428 (Contributed by Thierry Arnoux, 29-Nov-2017.)
⊢ Ⅎxφ    &   ⊢ (((φ ∧ x ∈ A) ∧ ψ) → χ)    &   ⊢ (φ → ∃x ∈ A ψ)    ⇒   ⊢ (φ → χ)
Theorem	r19.29a 2432*	A commonly used pattern based on r19.29 2428 (Contributed by Thierry Arnoux, 22-Nov-2017.)
⊢ (((φ ∧ x ∈ A) ∧ ψ) → χ)    &   ⊢ (φ → ∃x ∈ A ψ)    ⇒   ⊢ (φ → χ)
Theorem	r19.29d2r 2433	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	r19.29_2a 2434*	A commonly used pattern based on r19.29 2428, 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 2435	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 2436*	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 2437. (Contributed by Jim Kingdon, 19-Aug-2018.)
⊢ ((φ ∨ ∀x ∈ A ψ) → ∀x ∈ A (φ ∨ ψ))
Theorem	r19.32vdc 2437*	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 2438	Restricted quantifier version of 19.35-1 1497. (Contributed by Jim Kingdon, 4-Jun-2018.)
⊢ (∃x ∈ A (φ → ψ) → (∀x ∈ A φ → ∃x ∈ A ψ))
Theorem	r19.36av 2439*	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 2440	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 2441*	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 2442	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 2443	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 2444*	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 2445*	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 2446	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 2447*	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 2448*	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 2449*	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 2450*	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 2451*	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 2452*	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 2453*	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 2454*	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 2455	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 2456*	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 2457*	Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.)
⊢ (∃x ∈ A ∃y ∈ B (φ ∧ ψ) ↔ (∃x ∈ A φ ∧ ∃y ∈ B ψ))
Theorem	3reeanv 2458*	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 2459	x is not free in ∃!x ∈ Aφ. (Contributed by NM, 19-Mar-1997.)
⊢ Ⅎx∃!x ∈ A φ
Theorem	nfrmo1 2460	x is not free in ∃*x ∈ Aφ. (Contributed by NM, 16-Jun-2017.)
⊢ Ⅎx∃*x ∈ A φ
Theorem	nfreudxy 2461*	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 2462*	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 2463	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 2464*	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 2465	Equivalent wff's correspond to equal restricted class abstractions. Closed theorem form of rabbidva 2526. (Contributed by NM, 25-Nov-2013.)
⊢ (∀x ∈ A (ψ ↔ χ) ↔ {x ∈ A ∣ ψ} = {x ∈ A ∣ χ})
Theorem	rabswap 2466	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 2467	The abstraction variable in a restricted class abstraction isn't free. (Contributed by NM, 19-Mar-1997.)
⊢ Ⅎx{x ∈ A ∣ φ}
Theorem	nfrabxy 2468*	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 2469	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 2470*	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 2471*	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 2472	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 2473	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 2474	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 2475*	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 2476*	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 2477	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 2478	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 2479	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 2480	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 2481	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 2482	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 2483*	Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ))
Theorem	rexeq 2484*	Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ))
Theorem	reueq1 2485*	Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
⊢ (A = B → (∃!x ∈ A φ ↔ ∃!x ∈ B φ))
Theorem	rmoeq1 2486*	Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (A = B → (∃*x ∈ A φ ↔ ∃*x ∈ B φ))
Theorem	raleqi 2487*	Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ A = B    ⇒   ⊢ (∀x ∈ A φ ↔ ∀x ∈ B φ)
Theorem	rexeqi 2488*	Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
⊢ A = B    ⇒   ⊢ (∃x ∈ A φ ↔ ∃x ∈ B φ)
Theorem	raleqdv 2489*	Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
⊢ (φ → A = B)    ⇒   ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ B ψ))
Theorem	rexeqdv 2490*	Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
⊢ (φ → A = B)    ⇒   ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B ψ))
Theorem	raleqbi1dv 2491*	Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
⊢ (A = B → (φ ↔ ψ))    ⇒   ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B ψ))
Theorem	rexeqbi1dv 2492*	Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
⊢ (A = B → (φ ↔ ψ))    ⇒   ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B ψ))
Theorem	reueqd 2493*	Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
⊢ (A = B → (φ ↔ ψ))    ⇒   ⊢ (A = B → (∃!x ∈ A φ ↔ ∃!x ∈ B ψ))
Theorem	rmoeqd 2494*	Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
⊢ (A = B → (φ ↔ ψ))    ⇒   ⊢ (A = B → (∃*x ∈ A φ ↔ ∃*x ∈ B ψ))
Theorem	raleqbidv 2495*	Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
⊢ (φ → A = B)    &   ⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ B χ))
Theorem	rexeqbidv 2496*	Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
⊢ (φ → A = B)    &   ⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B χ))
Theorem	raleqbidva 2497*	Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ (φ → A = B)    &   ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))    ⇒   ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ B χ))
Theorem	rexeqbidva 2498*	Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
⊢ (φ → A = B)    &   ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ B χ))
Theorem	mormo 2499	Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
⊢ (∃*xφ → ∃*x ∈ A φ)
Theorem	reu5 2500	Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∃*x ∈ A φ))