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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  =  B  -> 
 ph )   =>    |-  ( -.  ph  ->  A  =/=  B )
 
Theoremnecon1aidc 2302 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
 |-  (DECID 
 ph  ->  ( -.  ph  ->  A  =  B ) )   =>    |-  (DECID 
 ph  ->  ( A  =/=  B 
 ->  ph ) )
 
Theoremnecon1bidc 2303 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
 |-  (DECID  A  =  B  ->  ( A  =/=  B  ->  ph ) )   =>    |-  (DECID  A  =  B  ->  ( -.  ph  ->  A  =  B ) )
 
Theoremnecon1idc 2304 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  ( A  =/=  B  ->  C  =  D )   =>    |-  (DECID  A  =  B  ->  ( C  =/=  D  ->  A  =  B ) )
 
Theoremnecon2ai 2305 Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
 |-  ( A  =  B  ->  -.  ph )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremnecon2bi 2306 Contrapositive inference for inequality. (Contributed by NM, 1-Apr-2007.)
 |-  ( ph  ->  A  =/=  B )   =>    |-  ( A  =  B  ->  -.  ph )
 
Theoremnecon2i 2307 Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.)
 |-  ( A  =  B  ->  C  =/=  D )   =>    |-  ( C  =  D  ->  A  =/=  B )
 
Theoremnecon2ad 2308 Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.)
 |-  ( ph  ->  ( A  =  B  ->  -. 
 ps ) )   =>    |-  ( ph  ->  ( ps  ->  A  =/=  B ) )
 
Theoremnecon2bd 2309 Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
 |-  ( ph  ->  ( ps  ->  A  =/=  B ) )   =>    |-  ( ph  ->  ( A  =  B  ->  -. 
 ps ) )
 
Theoremnecon2d 2310 Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)
 |-  ( ph  ->  ( A  =  B  ->  C  =/=  D ) )   =>    |-  ( ph  ->  ( C  =  D  ->  A  =/=  B ) )
 
Theoremnecon1abiidc 2311 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  (DECID 
 ph  ->  ( -.  ph  <->  A  =  B ) )   =>    |-  (DECID 
 ph  ->  ( A  =/=  B  <->  ph ) )
 
Theoremnecon1bbiidc 2312 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  (DECID  A  =  B  ->  ( A  =/=  B  <->  ph ) )   =>    |-  (DECID  A  =  B  ->  ( -.  ph  <->  A  =  B ) )
 
Theoremnecon1abiddc 2313 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  <->  A  =  B ) ) )   =>    |-  ( ph  ->  (DECID  ps 
 ->  ( A  =/=  B  <->  ps ) ) )
 
Theoremnecon1bbiddc 2314 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  ( ph  ->  (DECID  A  =  B  ->  ( A  =/=  B  <->  ps ) ) )   =>    |-  ( ph  ->  (DECID  A  =  B  ->  ( -.  ps  <->  A  =  B ) ) )
 
Theoremnecon2abiidc 2315 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  (DECID 
 ph  ->  ( A  =  B 
 <->  -.  ph ) )   =>    |-  (DECID 
 ph  ->  ( ph  <->  A  =/=  B ) )
 
Theoremnecon2bbii 2316 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  (DECID  A  =  B  ->  (
 ph 
 <->  A  =/=  B ) )   =>    |-  (DECID  A  =  B  ->  ( A  =  B  <->  -.  ph ) )
 
Theoremnecon2abiddc 2317 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  ( ph  ->  (DECID  ps  ->  ( A  =  B  <->  -. 
 ps ) ) )   =>    |-  ( ph  ->  (DECID  ps  ->  ( ps  <->  A  =/=  B ) ) )
 
Theoremnecon2bbiddc 2318 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  ( ph  ->  (DECID  A  =  B  ->  ( ps  <->  A  =/=  B ) ) )   =>    |-  ( ph  ->  (DECID  A  =  B  ->  ( A  =  B  <->  -.  ps ) ) )
 
Theoremnecon4aidc 2319 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
 |-  (DECID  A  =  B  ->  ( A  =/=  B  ->  -.  ph ) )   =>    |-  (DECID  A  =  B  ->  (
 ph  ->  A  =  B ) )
 
Theoremnecon4idc 2320 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 ) )
 
Theoremnecon4addc 2321 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
 |-  ( ph  ->  (DECID  A  =  B  ->  ( A  =/=  B  ->  -.  ps ) ) )   =>    |-  ( ph  ->  (DECID  A  =  B  ->  ( ps  ->  A  =  B ) ) )
 
Theoremnecon4bddc 2322 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
 |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  ->  A  =/=  B ) ) )   =>    |-  ( ph  ->  (DECID  ps  ->  ( A  =  B  ->  ps ) ) )
 
Theoremnecon4ddc 2323 Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
 |-  ( ph  ->  (DECID  A  =  B  ->  ( A  =/=  B  ->  C  =/=  D ) ) )   =>    |-  ( ph  ->  (DECID  A  =  B  ->  ( C  =  D  ->  A  =  B ) ) )
 
Theoremnecon4abiddc 2324 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 18-May-2018.)
 |-  ( ph  ->  (DECID  A  =  B  ->  (DECID  ps  ->  ( A  =/=  B  <->  -.  ps ) ) ) )   =>    |-  ( ph  ->  (DECID  A  =  B  ->  (DECID  ps  ->  ( A  =  B  <->  ps ) ) ) )
 
Theoremnecon4bbiddc 2325 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
 |-  ( ph  ->  (DECID  ps  ->  (DECID  A  =  B  ->  ( -.  ps  <->  A  =/=  B ) ) ) )   =>    |-  ( ph  ->  (DECID  ps 
 ->  (DECID  A  =  B  ->  ( ps  <->  A  =  B ) ) ) )
 
Theoremnecon4biddc 2326 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
 |-  ( ph  ->  (DECID  A  =  B  ->  (DECID  C  =  D  ->  ( A  =/=  B  <->  C  =/=  D ) ) ) )   =>    |-  ( ph  ->  (DECID  A  =  B  ->  (DECID  C  =  D  ->  ( A  =  B 
 <->  C  =  D ) ) ) )
 
Theoremnecon1addc 2327 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
 |-  ( ph  ->  (DECID  ps  ->  ( -.  ps  ->  A  =  B ) ) )   =>    |-  ( ph  ->  (DECID  ps  ->  ( A  =/=  B  ->  ps ) ) )
 
Theoremnecon1bddc 2328 Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
 |-  ( ph  ->  (DECID  A  =  B  ->  ( A  =/=  B  ->  ps )
 ) )   =>    |-  ( ph  ->  (DECID  A  =  B  ->  ( -. 
 ps  ->  A  =  B ) ) )
 
Theoremnecon1ddc 2329 Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
 |-  ( ph  ->  (DECID  A  =  B  ->  ( A  =/=  B  ->  C  =  D ) ) )   =>    |-  ( ph  ->  (DECID  A  =  B  ->  ( C  =/=  D 
 ->  A  =  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.)
 |-  ( ph  ->  -.  A  =  B )   =>    |-  ( ph  ->  A  =/=  B )
 
Theoremnebidc 2331 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 ) ) ) )
 
Theorempm13.18 2332 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( ( A  =  B  /\  A  =/=  C )  ->  B  =/=  C )
 
Theorempm13.181 2333 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( ( A  =  B  /\  B  =/=  C )  ->  A  =/=  C )
 
Theorempm2.21ddne 2334 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ps )
 
Theoremnecom 2335 Commutation of inequality. (Contributed by NM, 14-May-1999.)
 |-  ( A  =/=  B  <->  B  =/=  A )
 
Theoremnecomi 2336 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
 |-  A  =/=  B   =>    |-  B  =/=  A
 
Theoremnecomd 2337 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
 |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  B  =/=  A )
 
Theoremneanior 2338 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
 |-  ( ( A  =/=  B 
 /\  C  =/=  D ) 
 <->  -.  ( A  =  B  \/  C  =  D ) )
 
Theoremne3anior 2339 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 )
 )
 
Theoremnemtbir 2340 An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
 |-  A  =/=  B   &    |-  ( ph 
 <->  A  =  B )   =>    |-  -.  ph
 
Theoremnelne1 2341 Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
 |-  ( ( A  e.  B  /\  -.  A  e.  C )  ->  B  =/=  C )
 
Theoremnelne2 2342 Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)
 |-  ( ( A  e.  C  /\  -.  B  e.  C )  ->  A  =/=  B )
 
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.)
 |-  ( -.  A  e.  B  ->  ( C  e.  B  ->  C  =/=  A ) )
 
Theoremnfne 2344 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/ x  A  =/=  B
 
Theoremnfned 2345 Bound-variable hypothesis builder for inequality. (Contributed by NM, 10-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/ x  A  =/=  B )
 
2.1.4.2  Negated membership
 
Syntaxwnel 2346 Extend wff notation to include negated membership.
 wff  A  e/  B
 
Definitiondf-nel 2347 Define negated membership. (Contributed by NM, 7-Aug-1994.)
 |-  ( A  e/  B  <->  -.  A  e.  B )
 
Theoremneli 2348 Inference associated with df-nel 2347. (Contributed by BJ, 7-Jul-2018.)
 |-  A  e/  B   =>    |-  -.  A  e.  B
 
Theoremnelir 2349 Inference associated with df-nel 2347. (Contributed by BJ, 7-Jul-2018.)
 |- 
 -.  A  e.  B   =>    |-  A  e/  B
 
Theoremneleq1 2350 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
 |-  ( A  =  B  ->  ( A  e/  C  <->  B 
 e/  C ) )
 
Theoremneleq2 2351 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
 |-  ( A  =  B  ->  ( C  e/  A  <->  C 
 e/  B ) )
 
Theoremneleq12d 2352 Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( A  e/  C  <->  B  e/  D ) )
 
Theoremnfnel 2353 Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/ x  A  e/  B
 
Theoremnfneld 2354 Bound-variable hypothesis builder for negated membership. (Contributed by David Abernethy, 26-Jun-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/ x  A  e/  B )
 
2.1.5  Restricted quantification
 
Syntaxwral 2355 Extend wff notation to include restricted universal quantification.
 wff  A. x  e.  A  ph
 
Syntaxwrex 2356 Extend wff notation to include restricted existential quantification.
 wff  E. x  e.  A  ph
 
Syntaxwreu 2357 Extend wff notation to include restricted existential uniqueness.
 wff  E! x  e.  A  ph
 
Syntaxwrmo 2358 Extend wff notation to include restricted "at most one."
 wff  E* x  e.  A  ph
 
Syntaxcrab 2359 Extend class notation to include the restricted class abstraction (class builder).
 class  { x  e.  A  |  ph }
 
Definitiondf-ral 2360 Define restricted universal quantification. Special case of Definition 4.15(3) of [TakeutiZaring] p. 22. (Contributed by NM, 19-Aug-1993.)
 |-  ( A. x  e.  A  ph  <->  A. x ( x  e.  A  ->  ph )
 )
 
Definitiondf-rex 2361 Define restricted existential quantification. Special case of Definition 4.15(4) of [TakeutiZaring] p. 22. (Contributed by NM, 30-Aug-1993.)
 |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
 )
 
Definitiondf-reu 2362 Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
 |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
 )
 
Definitiondf-rmo 2363 Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
 |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
 )
 
Definitiondf-rab 2364 Define a restricted class abstraction (class builder), which is the class of all  x in  A such that  ph is true. Definition of [TakeutiZaring] p. 20. (Contributed by NM, 22-Nov-1994.)
 |- 
 { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
 
Theoremralnex 2365 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
 |-  ( A. x  e.  A  -.  ph  <->  -.  E. x  e.  A  ph )
 
Theoremrexnalim 2366 Relationship between restricted universal and existential quantifiers. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 17-Aug-2018.)
 |-  ( E. x  e.  A  -.  ph  ->  -. 
 A. x  e.  A  ph )
 
Theoremdfrex2dc 2367 Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 29-Jun-2022.)
 |-  (DECID 
 E. x  e.  A  ph 
 ->  ( E. x  e.  A  ph  <->  -.  A. x  e.  A  -.  ph )
 )
 
Theoremralexim 2368 Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
 |-  ( A. x  e.  A  ph  ->  -.  E. x  e.  A  -.  ph )
 
Theoremrexalim 2369 Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)
 |-  ( E. x  e.  A  ph  ->  -.  A. x  e.  A  -.  ph )
 
Theoremralbida 2370 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
 |- 
 F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  A  ch ) )
 
Theoremrexbida 2371 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)
 |- 
 F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  A  ch ) )
 
Theoremralbidva 2372* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 4-Mar-1997.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  A  ch ) )
 
Theoremrexbidva 2373* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 9-Mar-1997.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  A  ch ) )
 
Theoremralbid 2374 Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  A  ch ) )
 
Theoremrexbid 2375 Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  A  ch ) )
 
Theoremralbidv 2376* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  A  ch ) )
 
Theoremrexbidv 2377* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 20-Nov-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  A  ch ) )
 
Theoremralbidv2 2378* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 6-Apr-1997.)
 |-  ( ph  ->  (
 ( x  e.  A  ->  ps )  <->  ( x  e.  B  ->  ch )
 ) )   =>    |-  ( ph  ->  ( A. x  e.  A  ps 
 <-> 
 A. x  e.  B  ch ) )
 
Theoremrexbidv2 2379* Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 22-May-1999.)
 |-  ( ph  ->  (
 ( x  e.  A  /\  ps )  <->  ( x  e.  B  /\  ch )
 ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  B  ch ) )
 
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.)
 |-  ( ph  <->  ps )   =>    |-  ( A. x  e.  A  ph  <->  A. x  e.  A  ps )
 
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.)
 |-  ( ph  <->  ps )   =>    |-  ( E. x  e.  A  ph  <->  E. x  e.  A  ps )
 
Theorem2ralbii 2382 Inference adding two restricted universal quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
 |-  ( ph  <->  ps )   =>    |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. x  e.  A  A. y  e.  B  ps )
 
Theorem2rexbii 2383 Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 11-Nov-1995.)
 |-  ( ph  <->  ps )   =>    |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x  e.  A  E. y  e.  B  ps )
 
Theoremralbii2 2384 Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)
 |-  ( ( x  e.  A  ->  ph )  <->  ( x  e.  B  ->  ps )
 )   =>    |-  ( A. x  e.  A  ph  <->  A. x  e.  B  ps )
 
Theoremrexbii2 2385 Inference adding different restricted existential quantifiers to each side of an equivalence. (Contributed by NM, 4-Feb-2004.)
 |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ps ) )   =>    |-  ( E. x  e.  A  ph  <->  E. x  e.  B  ps )
 
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   &    |-  ( ps 
 <->  ch )   =>    |-  ( A. x  e.  A  ps  <->  A. x  e.  B  ch )
 
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   &    |-  ( ps 
 <->  ch )   =>    |-  ( E. x  e.  A  ps  <->  E. x  e.  B  ch )
 
Theoremralbiia 2388 Inference adding restricted universal quantifier to both sides of an equivalence. (Contributed by NM, 26-Nov-2000.)
 |-  ( x  e.  A  ->  ( ph  <->  ps ) )   =>    |-  ( A. x  e.  A  ph  <->  A. x  e.  A  ps )
 
Theoremrexbiia 2389 Inference adding restricted existential quantifier to both sides of an equivalence. (Contributed by NM, 26-Oct-1999.)
 |-  ( x  e.  A  ->  ( ph  <->  ps ) )   =>    |-  ( E. x  e.  A  ph  <->  E. x  e.  A  ps )
 
Theorem2rexbiia 2390* Inference adding two restricted existential quantifiers to both sides of an equivalence. (Contributed by NM, 1-Aug-2004.)
 |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x  e.  A  E. y  e.  B  ps )
 
Theoremr2alf 2391* Double restricted universal quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ y A   =>    |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. x A. y
 ( ( x  e.  A  /\  y  e.  B )  ->  ph )
 )
 
Theoremr2exf 2392* Double restricted existential quantification. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |-  F/_ y A   =>    |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x E. y
 ( ( x  e.  A  /\  y  e.  B )  /\  ph )
 )
 
Theoremr2al 2393* Double restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
 |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. x A. y
 ( ( x  e.  A  /\  y  e.  B )  ->  ph )
 )
 
Theoremr2ex 2394* Double restricted existential quantification. (Contributed by NM, 11-Nov-1995.)
 |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. x E. y
 ( ( x  e.  A  /\  y  e.  B )  /\  ph )
 )
 
Theorem2ralbida 2395* Formula-building rule for restricted universal quantifier (deduction rule). (Contributed by NM, 24-Feb-2004.)
 |- 
 F/ x ph   &    |-  F/ y ph   &    |-  (
 ( ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  A. y  e.  B  ps  <->  A. x  e.  A  A. y  e.  B  ch ) )
 
Theorem2ralbidva 2396* Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( A. x  e.  A  A. y  e.  B  ps  <->  A. x  e.  A  A. y  e.  B  ch ) )
 
Theorem2rexbidva 2397* Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 15-Dec-2004.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  E. y  e.  B  ps 
 <-> 
 E. x  e.  A  E. y  e.  B  ch ) )
 
Theorem2ralbidv 2398* Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  A. y  e.  B  ps  <->  A. x  e.  A  A. y  e.  B  ch ) )
 
Theorem2rexbidv 2399* Formula-building rule for restricted existential quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  E. y  e.  B  ps 
 <-> 
 E. x  e.  A  E. y  e.  B  ch ) )
 
Theoremrexralbidv 2400* Formula-building rule for restricted quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  A. y  e.  B  ps  <->  E. x  e.  A  A. y  e.  B  ch ) )
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