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Theorem List for Intuitionistic Logic Explorer - 2301-2400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnecon3bi 2301 Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
(𝐴 = 𝐵𝜑)       𝜑𝐴𝐵)

Theoremnecon1aidc 2302 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
(DECID 𝜑 → (¬ 𝜑𝐴 = 𝐵))       (DECID 𝜑 → (𝐴𝐵𝜑))

Theoremnecon1bidc 2303 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
(DECID 𝐴 = 𝐵 → (𝐴𝐵𝜑))       (DECID 𝐴 = 𝐵 → (¬ 𝜑𝐴 = 𝐵))

Theoremnecon1idc 2304 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(𝐴𝐵𝐶 = 𝐷)       (DECID 𝐴 = 𝐵 → (𝐶𝐷𝐴 = 𝐵))

Theoremnecon2ai 2305 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
(𝐴 = 𝐵 → ¬ 𝜑)       (𝜑𝐴𝐵)

Theoremnecon2bi 2306 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
(𝜑𝐴𝐵)       (𝐴 = 𝐵 → ¬ 𝜑)

Theoremnecon2i 2307 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
(𝐴 = 𝐵𝐶𝐷)       (𝐶 = 𝐷𝐴𝐵)

Theoremnecon2ad 2308 Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
(𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))       (𝜑 → (𝜓𝐴𝐵))

Theoremnecon2bd 2309 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
(𝜑 → (𝜓𝐴𝐵))       (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓))

Theoremnecon2d 2310 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
(𝜑 → (𝐴 = 𝐵𝐶𝐷))       (𝜑 → (𝐶 = 𝐷𝐴𝐵))

Theoremnecon1abiidc 2311 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID 𝜑 → (¬ 𝜑𝐴 = 𝐵))       (DECID 𝜑 → (𝐴𝐵𝜑))

Theoremnecon1bbiidc 2312 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID 𝐴 = 𝐵 → (𝐴𝐵𝜑))       (DECID 𝐴 = 𝐵 → (¬ 𝜑𝐴 = 𝐵))

Theoremnecon1abiddc 2313 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴 = 𝐵)))       (𝜑 → (DECID 𝜓 → (𝐴𝐵𝜓)))

Theoremnecon1bbiddc 2314 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜓)))       (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓𝐴 = 𝐵)))

Theoremnecon2abiidc 2315 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID 𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜑))       (DECID 𝜑 → (𝜑𝐴𝐵))

Theoremnecon2bbii 2316 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID 𝐴 = 𝐵 → (𝜑𝐴𝐵))       (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜑))

Theoremnecon2abiddc 2317 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))       (𝜑 → (DECID 𝜓 → (𝜓𝐴𝐵)))

Theoremnecon2bbiddc 2318 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (𝜓𝐴𝐵)))       (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))

Theoremnecon4aidc 2319 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID 𝐴 = 𝐵 → (𝐴𝐵 → ¬ 𝜑))       (DECID 𝐴 = 𝐵 → (𝜑𝐴 = 𝐵))

Theoremnecon4idc 2320 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
(DECID 𝐴 = 𝐵 → (𝐴𝐵𝐶𝐷))       (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷𝐴 = 𝐵))

Theoremnecon4addc 2321 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵 → ¬ 𝜓)))       (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓𝐴 = 𝐵)))

Theoremnecon4bddc 2322 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
(𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴𝐵)))       (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵𝜓)))

Theoremnecon4ddc 2323 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝐶𝐷)))       (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶 = 𝐷𝐴 = 𝐵)))

Theoremnecon4abiddc 2324 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 18-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴𝐵 ↔ ¬ 𝜓))))       (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 = 𝐵𝜓))))

Theoremnecon4bbiddc 2325 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (¬ 𝜓𝐴𝐵))))       (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝜓𝐴 = 𝐵))))

Theoremnecon4biddc 2326 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴𝐵𝐶𝐷))))       (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → (𝐴 = 𝐵𝐶 = 𝐷))))

Theoremnecon1addc 2327 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴 = 𝐵)))       (𝜑 → (DECID 𝜓 → (𝐴𝐵𝜓)))

Theoremnecon1bddc 2328 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜓)))       (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓𝐴 = 𝐵)))

Theoremnecon1ddc 2329 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝐶 = 𝐷)))       (𝜑 → (DECID 𝐴 = 𝐵 → (𝐶𝐷𝐴 = 𝐵)))

Theoremneneqad 2330 If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2272. One-way deduction form of df-ne 2252. (Contributed by David Moews, 28-Feb-2017.)
(𝜑 → ¬ 𝐴 = 𝐵)       (𝜑𝐴𝐵)

Theoremnebidc 2331 Contraposition law for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
(DECID 𝐴 = 𝐵 → (DECID 𝐶 = 𝐷 → ((𝐴 = 𝐵𝐶 = 𝐷) ↔ (𝐴𝐵𝐶𝐷))))

Theorempm13.18 2332 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝐴 = 𝐵𝐴𝐶) → 𝐵𝐶)

Theorempm13.181 2333 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
((𝐴 = 𝐵𝐵𝐶) → 𝐴𝐶)

Theorempm2.21ddne 2334 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐴𝐵)       (𝜑𝜓)

Theoremnecom 2335 Commutation of inequality. (Contributed by NM, 14-May-1999.)
(𝐴𝐵𝐵𝐴)

Theoremnecomi 2336 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
𝐴𝐵       𝐵𝐴

Theoremnecomd 2337 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
(𝜑𝐴𝐵)       (𝜑𝐵𝐴)

Theoremneanior 2338 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
((𝐴𝐵𝐶𝐷) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷))

Theoremne3anior 2339 A De Morgan's law for inequality. (Contributed by NM, 30-Sep-2013.) (Proof rewritten by Jim Kingdon, 19-May-2018.)
((𝐴𝐵𝐶𝐷𝐸𝐹) ↔ ¬ (𝐴 = 𝐵𝐶 = 𝐷𝐸 = 𝐹))

Theoremnemtbir 2340 An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
𝐴𝐵    &   (𝜑𝐴 = 𝐵)        ¬ 𝜑

Theoremnelne1 2341 Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)

Theoremnelne2 2342 Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)
((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)

Theoremnelelne 2343 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.)
𝐴𝐵 → (𝐶𝐵𝐶𝐴))

Theoremnfne 2344 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵

Theoremnfned 2345 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)

2.1.4.2  Negated membership

Syntaxwnel 2346 Extend wff notation to include negated membership.
wff 𝐴𝐵

Definitiondf-nel 2347 Define negated membership. (Contributed by NM, 7-Aug-1994.)
(𝐴𝐵 ↔ ¬ 𝐴𝐵)

Theoremneli 2348 Inference associated with df-nel 2347. (Contributed by BJ, 7-Jul-2018.)
𝐴𝐵        ¬ 𝐴𝐵

Theoremnelir 2349 Inference associated with df-nel 2347. (Contributed by BJ, 7-Jul-2018.)
¬ 𝐴𝐵       𝐴𝐵

Theoremneleq1 2350 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
(𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))

Theoremneleq2 2351 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
(𝐴 = 𝐵 → (𝐶𝐴𝐶𝐵))

Theoremneleq12d 2352 Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴𝐶𝐵𝐷))

Theoremnfnel 2353 Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
𝑥𝐴    &   𝑥𝐵       𝑥 𝐴𝐵

Theoremnfneld 2354 Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
(𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑 → Ⅎ𝑥 𝐴𝐵)

2.1.5  Restricted quantification

Syntaxwral 2355 Extend wff notation to include restricted universal quantification.
wff 𝑥𝐴 𝜑

Syntaxwrex 2356 Extend wff notation to include restricted existential quantification.
wff 𝑥𝐴 𝜑

Syntaxwreu 2357 Extend wff notation to include restricted existential uniqueness.
wff ∃!𝑥𝐴 𝜑

Syntaxwrmo 2358 Extend wff notation to include restricted "at most one."
wff ∃*𝑥𝐴 𝜑

Syntaxcrab 2359 Extend class notation to include the restricted class abstraction (class builder).
class {𝑥𝐴𝜑}

Definitiondf-ral 2360 Define restricted universal quantification. Special case of Definition 4.15(3) of [TakeutiZaring] p. 22. (Contributed by NM, 19-Aug-1993.)
(∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))

Definitiondf-rex 2361 Define restricted existential quantification. Special case of Definition 4.15(4) of [TakeutiZaring] p. 22. (Contributed by NM, 30-Aug-1993.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))

Definitiondf-reu 2362 Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
(∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))

Definitiondf-rmo 2363 Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
(∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))

Definitiondf-rab 2364 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.)
{𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}

Theoremralnex 2365 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
(∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)

Theoremrexnalim 2366 Relationship between restricted universal and existential quantifiers. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 17-Aug-2018.)
(∃𝑥𝐴 ¬ 𝜑 → ¬ ∀𝑥𝐴 𝜑)

Theoremdfrex2dc 2367 Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 29-Jun-2022.)
(DECID𝑥𝐴 𝜑 → (∃𝑥𝐴 𝜑 ↔ ¬ ∀𝑥𝐴 ¬ 𝜑))

Theoremralexim 2368 Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
(∀𝑥𝐴 𝜑 → ¬ ∃𝑥𝐴 ¬ 𝜑)

Theoremrexalim 2369 Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
(∃𝑥𝐴 𝜑 → ¬ ∀𝑥𝐴 ¬ 𝜑)

Theoremralbida 2370 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))

Theoremrexbida 2371 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
𝑥𝜑    &   ((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))

Theoremralbidva 2372* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 4-Mar-1997.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))

Theoremrexbidva 2373* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 9-Mar-1997.)
((𝜑𝑥𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))

Theoremralbid 2374 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))

Theoremrexbid 2375 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
𝑥𝜑    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))

Theoremralbidv 2376* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐴 𝜒))

Theoremrexbidv 2377* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐴 𝜒))

Theoremralbidv2 2378* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Apr-1997.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒))

Theoremrexbidv2 2379* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 22-May-1999.)
(𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐵𝜒)))       (𝜑 → (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒))

Theoremralbii 2380 Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario Carneiro, 17-Oct-2016.)
(𝜑𝜓)       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓)

Theoremrexbii 2381 Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 23-Nov-1994.) (Revised by Mario Carneiro, 17-Oct-2016.)
(𝜑𝜓)       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓)

Theorem2ralbii 2382 Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
(𝜑𝜓)       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝐴𝑦𝐵 𝜓)

Theorem2rexbii 2383 Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.)
(𝜑𝜓)       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)

Theoremralbii2 2384 Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)
((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓)

Theoremrexbii2 2385 Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)
((𝑥𝐴𝜑) ↔ (𝑥𝐵𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐵 𝜓)

Theoremraleqbii 2386 Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
𝐴 = 𝐵    &   (𝜓𝜒)       (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜒)

Theoremrexeqbii 2387 Equality deduction for restricted existential quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.)
𝐴 = 𝐵    &   (𝜓𝜒)       (∃𝑥𝐴 𝜓 ↔ ∃𝑥𝐵 𝜒)

Theoremralbiia 2388 Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)
(𝑥𝐴 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜓)

Theoremrexbiia 2389 Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)
(𝑥𝐴 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 𝜓)

Theorem2rexbiia 2390* Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
((𝑥𝐴𝑦𝐵) → (𝜑𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑦𝐵 𝜓)

Theoremr2alf 2391* Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))

Theoremr2exf 2392* Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
𝑦𝐴       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))

Theoremr2al 2393* Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
(∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐵) → 𝜑))

Theoremr2ex 2394* Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ 𝜑))

Theorem2ralbida 2395* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 24-Feb-2004.)
𝑥𝜑    &   𝑦𝜑    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))

Theorem2ralbidva 2396* Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))

Theorem2rexbidva 2397* Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 15-Dec-2004.)
((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))

Theorem2ralbidv 2398* Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝐴𝑦𝐵 𝜓 ↔ ∀𝑥𝐴𝑦𝐵 𝜒))

Theorem2rexbidv 2399* Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))

Theoremrexralbidv 2400* Formula-building rule for restricted quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝐴𝑦𝐵 𝜓 ↔ ∃𝑥𝐴𝑦𝐵 𝜒))

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