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	necon2bbiidc 2401	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (𝜑 ↔ 𝐴 ≠ 𝐵))    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜑))
Theorem	necon2abiddc 2402	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (𝜓 ↔ 𝐴 ≠ 𝐵)))
Theorem	necon2bbiddc 2403	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓 ↔ 𝐴 ≠ 𝐵)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))
Theorem	necon4aidc 2404	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ¬ 𝜑))    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (𝜑 → 𝐴 = 𝐵))
Theorem	necon4idc 2405	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷))    ⇒   ⊢ (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷 → 𝐴 = 𝐵))
Theorem	necon4addc 2406	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → ¬ 𝜓)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓 → 𝐴 = 𝐵)))
Theorem	necon4bddc 2407	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 ≠ 𝐵)))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 → 𝜓)))
Theorem	necon4ddc 2408	Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷 → 𝐴 = 𝐵)))
Theorem	necon4abiddc 2409	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 18-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓))))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ 𝜓))))
Theorem	necon4bbiddc 2410	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵))))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝜓 ↔ 𝐴 = 𝐵))))
Theorem	necon4biddc 2411	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))))
Theorem	necon1addc 2412	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝐴 = 𝐵)))    ⇒   ⊢ (𝜑 → (DECID 𝜓 → (𝐴 ≠ 𝐵 → 𝜓)))
Theorem	necon1bddc 2413	Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝜓)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓 → 𝐴 = 𝐵)))
Theorem	necon1ddc 2414	Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 → 𝐶 = 𝐷)))    ⇒   ⊢ (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵)))
Theorem	neneqad 2415	If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2357. One-way deduction form of df-ne 2337. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → ¬ 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≠ 𝐵)
Theorem	nebidc 2416	Contraposition law for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
⊢ (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → ((𝐴 = 𝐵 ↔ 𝐶 = 𝐷) ↔ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷))))
Theorem	pm13.18 2417	Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ ((𝐴 = 𝐵 ∧ 𝐴 ≠ 𝐶) → 𝐵 ≠ 𝐶)
Theorem	pm13.181 2418	Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
⊢ ((𝐴 = 𝐵 ∧ 𝐵 ≠ 𝐶) → 𝐴 ≠ 𝐶)
Theorem	pm2.21ddne 2419	A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	necom 2420	Commutation of inequality. (Contributed by NM, 14-May-1999.)
⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
Theorem	necomi 2421	Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
⊢ 𝐴 ≠ 𝐵    ⇒   ⊢ 𝐵 ≠ 𝐴
Theorem	necomd 2422	Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
⊢ (𝜑 → 𝐴 ≠ 𝐵)    ⇒   ⊢ (𝜑 → 𝐵 ≠ 𝐴)
Theorem	neanior 2423	A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷))
Theorem	ne3anior 2424	A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.) (Proof rewritten by Jim Kingdon, 19-May-2018.)
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐷 ∧ 𝐸 ≠ 𝐹) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐶 = 𝐷 ∨ 𝐸 = 𝐹))
Theorem	nemtbir 2425	An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
⊢ 𝐴 ≠ 𝐵    &   ⊢ (𝜑 ↔ 𝐴 = 𝐵)    ⇒   ⊢ ¬ 𝜑
Theorem	nelne1 2426	Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐵 ≠ 𝐶)
Theorem	nelne2 2427	Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)
⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → 𝐴 ≠ 𝐵)
Theorem	nelelne 2428	Two classes are different if they don't belong to the same class. (Contributed by Rodolfo Medina, 17-Oct-2010.) (Proof shortened by AV, 10-May-2020.)
⊢ (¬ 𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝐶 ≠ 𝐴))
Theorem	nfne 2429	Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 ≠ 𝐵
Theorem	nfned 2430	Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴 ≠ 𝐵)
2.1.4.2  Negated membership
Syntax	wnel 2431	Extend wff notation to include negated membership.
wff 𝐴 ∉ 𝐵
Definition	df-nel 2432	Define negated membership. (Contributed by NM, 7-Aug-1994.)
⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
Theorem	neli 2433	Inference associated with df-nel 2432. (Contributed by BJ, 7-Jul-2018.)
⊢ 𝐴 ∉ 𝐵    ⇒   ⊢ ¬ 𝐴 ∈ 𝐵
Theorem	nelir 2434	Inference associated with df-nel 2432. (Contributed by BJ, 7-Jul-2018.)
⊢ ¬ 𝐴 ∈ 𝐵    ⇒   ⊢ 𝐴 ∉ 𝐵
Theorem	neleq1 2435	Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶))
Theorem	neleq2 2436	Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
⊢ (𝐴 = 𝐵 → (𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵))
Theorem	neleq12d 2437	Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷))
Theorem	nfnel 2438	Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 ∉ 𝐵
Theorem	nfneld 2439	Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∉ 𝐵)
Theorem	elnelne1 2440	Two classes are different if they don't contain the same element. (Contributed by AV, 28-Jan-2020.)
⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∉ 𝐶) → 𝐵 ≠ 𝐶)
Theorem	elnelne2 2441	Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∉ 𝐶) → 𝐴 ≠ 𝐵)
Theorem	nelcon3d 2442	Contrapositive law deduction for negated membership. (Contributed by AV, 28-Jan-2020.)
⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝐶 ∈ 𝐷))    ⇒   ⊢ (𝜑 → (𝐶 ∉ 𝐷 → 𝐴 ∉ 𝐵))
Theorem	elnelall 2443	A contradiction concerning membership implies anything. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
⊢ (𝐴 ∈ 𝐵 → (𝐴 ∉ 𝐵 → 𝜑))
2.1.5  Restricted quantification
Syntax	wral 2444	Extend wff notation to include restricted universal quantification.
wff ∀𝑥 ∈ 𝐴 𝜑
Syntax	wrex 2445	Extend wff notation to include restricted existential quantification.
wff ∃𝑥 ∈ 𝐴 𝜑
Syntax	wreu 2446	Extend wff notation to include restricted existential uniqueness.
wff ∃!𝑥 ∈ 𝐴 𝜑
Syntax	wrmo 2447	Extend wff notation to include restricted "at most one".
wff ∃*𝑥 ∈ 𝐴 𝜑
Syntax	crab 2448	Extend class notation to include the restricted class abstraction (class builder).
class {𝑥 ∈ 𝐴 ∣ 𝜑}
Definition	df-ral 2449	Define restricted universal quantification. Special case of Definition 4.15(3) of [TakeutiZaring] p. 22. (Contributed by NM, 19-Aug-1993.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
Definition	df-rex 2450	Define restricted existential quantification. Special case of Definition 4.15(4) of [TakeutiZaring] p. 22. (Contributed by NM, 30-Aug-1993.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Definition	df-reu 2451	Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Definition	df-rmo 2452	Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
Definition	df-rab 2453	Define a restricted class abstraction (class builder), which is the class of all 𝑥 in 𝐴 such that 𝜑 is true. Definition of [TakeutiZaring] p. 20. (Contributed by NM, 22-Nov-1994.)
⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
Theorem	ralnex 2454	Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
Theorem	rexnalim 2455	Relationship between restricted universal and existential quantifiers. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 17-Aug-2018.)
⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ ∀𝑥 ∈ 𝐴 𝜑)
Theorem	nnral 2456	The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1637. (Contributed by Jim Kingdon, 1-Aug-2024.)
⊢ (¬ ¬ ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑)
Theorem	dfrex2dc 2457	Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 29-Jun-2022.)
⊢ (DECID ∃𝑥 ∈ 𝐴 𝜑 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑))
Theorem	ralexim 2458	Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
Theorem	rexalim 2459	Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
⊢ (∃𝑥 ∈ 𝐴 𝜑 → ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
Theorem	ralbida 2460	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
Theorem	rexbida 2461	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 6-Oct-2003.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))
Theorem	ralbidva 2462*	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 4-Mar-1997.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
Theorem	rexbidva 2463*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 9-Mar-1997.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))
Theorem	ralbid 2464	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
Theorem	rexbid 2465	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))
Theorem	ralbidv 2466*	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 20-Nov-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒))
Theorem	rexbidv 2467*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 20-Nov-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒))
Theorem	ralbidv2 2468*	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Apr-1997.)
⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
Theorem	rexbidv2 2469*	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 22-May-1999.)
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
Theorem	ralbid2 2470	Formula-building rule for restricted universal quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
Theorem	rexbid2 2471	Formula-building rule for restricted existential quantifier (deduction form). (Contributed by BJ, 14-Jul-2024.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜒))
Theorem	ralbii 2472	Inference adding restricted universal 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	rexbii 2473	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 2474	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 2475	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 2476	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 2477	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 2478	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 2479	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 2480	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 2481	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 2482*	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 2483*	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 2484*	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 2485*	Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑))
Theorem	r2ex 2486*	Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)
⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))
Theorem	2ralbida 2487*	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 2488*	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 2489*	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 2490*	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 2491*	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 2492*	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 2493	A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
Theorem	risset 2494*	Two ways to say "𝐴 belongs to 𝐵". (Contributed by NM, 22-Nov-1994.)
⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐵 𝑥 = 𝐴)
Theorem	hbral 2495	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 2496	𝑥 is not free in ∀𝑥 ∈ 𝐴𝜑. (Contributed by NM, 18-Oct-1996.)
⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥∀𝑥 ∈ 𝐴 𝜑)
Theorem	nfra1 2497	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	nfraldw 2498*	Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfraldya 2501 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)
Theorem	nfraldxy 2499*	Old name for nfraldw 2498. (Contributed by Jim Kingdon, 29-May-2018.) (New usage is discouraged.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓)
Theorem	nfrexdxy 2500*	Not-free for restricted existential quantification where 𝑥 and 𝑦 are distinct. See nfrexdya 2502 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)