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Theorem List for Intuitionistic Logic Explorer - 2301-2400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempm13.18 2301 Theorem *13.18 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( ( A  =  B  /\  A  =/=  C )  ->  B  =/=  C )
 
Theorempm13.181 2302 Theorem *13.181 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( ( A  =  B  /\  B  =/=  C )  ->  A  =/=  C )
 
Theorempm2.21ddne 2303 A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  ps )
 
Theoremnecom 2304 Commutation of inequality. (Contributed by NM, 14-May-1999.)
 |-  ( A  =/=  B  <->  B  =/=  A )
 
Theoremnecomi 2305 Inference from commutative law for inequality. (Contributed by NM, 17-Oct-2012.)
 |-  A  =/=  B   =>    |-  B  =/=  A
 
Theoremnecomd 2306 Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.)
 |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  B  =/=  A )
 
Theoremneanior 2307 A De Morgan's law for inequality. (Contributed by NM, 18-May-2007.)
 |-  ( ( A  =/=  B 
 /\  C  =/=  D ) 
 <->  -.  ( A  =  B  \/  C  =  D ) )
 
Theoremne3anior 2308 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 2309 An inference from an inequality, related to modus tollens. (Contributed by NM, 13-Apr-2007.)
 |-  A  =/=  B   &    |-  ( ph 
 <->  A  =  B )   =>    |-  -.  ph
 
Theoremnelne1 2310 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 2311 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 )
 
Theoremnfne 2312 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 2313 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 2314 Extend wff notation to include negated membership.
 wff  A  e/  B
 
Definitiondf-nel 2315 Define negated membership. (Contributed by NM, 7-Aug-1994.)
 |-  ( A  e/  B  <->  -.  A  e.  B )
 
Theoremneli 2316 Inference associated with df-nel 2315. (Contributed by BJ, 7-Jul-2018.)
 |-  A  e/  B   =>    |-  -.  A  e.  B
 
Theoremnelir 2317 Inference associated with df-nel 2315. (Contributed by BJ, 7-Jul-2018.)
 |- 
 -.  A  e.  B   =>    |-  A  e/  B
 
Theoremneleq1 2318 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
 |-  ( A  =  B  ->  ( A  e/  C  <->  B 
 e/  C ) )
 
Theoremneleq2 2319 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
 |-  ( A  =  B  ->  ( C  e/  A  <->  C 
 e/  B ) )
 
Theoremneleq12d 2320 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 2321 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 2322 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 2323 Extend wff notation to include restricted universal quantification.
 wff  A. x  e.  A  ph
 
Syntaxwrex 2324 Extend wff notation to include restricted existential quantification.
 wff  E. x  e.  A  ph
 
Syntaxwreu 2325 Extend wff notation to include restricted existential uniqueness.
 wff  E! x  e.  A  ph
 
Syntaxwrmo 2326 Extend wff notation to include restricted "at most one."
 wff  E* x  e.  A  ph
 
Syntaxcrab 2327 Extend class notation to include the restricted class abstraction (class builder).
 class  { x  e.  A  |  ph }
 
Definitiondf-ral 2328 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 2329 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 2330 Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
 |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
 )
 
Definitiondf-rmo 2331 Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
 |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
 )
 
Definitiondf-rab 2332 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 2333 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
 |-  ( A. x  e.  A  -.  ph  <->  -.  E. x  e.  A  ph )
 
Theoremrexnalim 2334 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 )
 
Theoremralexim 2335 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 2336 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 2337 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 2338 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 2339* 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 2340* 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 2341 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 2342 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 2343* 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 2344* 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 2345* 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 2346* 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 2347 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 2348 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 2349 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 2350 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 2351 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 2352 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 2353 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 2354 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 2355 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 2356 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 2357* 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 2358* 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 2359* 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 2360* 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 2361* 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 2362* 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 2363* 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 2364* 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 2365* 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 2366* 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 2367* 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 ) )
 
Theoremralinexa 2368 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
 |-  ( A. x  e.  A  ( ph  ->  -. 
 ps )  <->  -.  E. x  e.  A  ( ph  /\  ps ) )
 
Theoremrisset 2369* Two ways to say " A belongs to  B." (Contributed by NM, 22-Nov-1994.)
 |-  ( A  e.  B  <->  E. x  e.  B  x  =  A )
 
Theoremhbral 2370 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.)
 |-  ( y  e.  A  ->  A. x  y  e.  A )   &    |-  ( ph  ->  A. x ph )   =>    |-  ( A. y  e.  A  ph  ->  A. x A. y  e.  A  ph )
 
Theoremhbra1 2371  x is not free in  A. x  e.  A ph. (Contributed by NM, 18-Oct-1996.)
 |-  ( A. x  e.  A  ph  ->  A. x A. x  e.  A  ph )
 
Theoremnfra1 2372  x is not free in  A. x  e.  A ph. (Contributed by NM, 18-Oct-1996.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ x A. x  e.  A  ph
 
Theoremnfraldxy 2373* Not-free for restricted universal quantification where  x and  y are distinct. See nfraldya 2375 for a version with  y and  A distinct instead. (Contributed by Jim Kingdon, 29-May-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y  e.  A  ps )
 
Theoremnfrexdxy 2374* Not-free for restricted existential quantification where  x and  y are distinct. See nfrexdya 2376 for a version with  y and  A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E. y  e.  A  ps )
 
Theoremnfraldya 2375* Not-free for restricted universal quantification where  y and  A are distinct. See nfraldxy 2373 for a version with  x and  y distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y  e.  A  ps )
 
Theoremnfrexdya 2376* Not-free for restricted existential quantification where  y and  A are distinct. See nfrexdxy 2374 for a version with  x and  y distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E. y  e.  A  ps )
 
Theoremnfralxy 2377* Not-free for restricted universal quantification where  x and  y are distinct. See nfralya 2379 for a version with  y and 
A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x A. y  e.  A  ph
 
Theoremnfrexxy 2378* Not-free for restricted existential quantification where  x and  y are distinct. See nfrexya 2380 for a version with  y and 
A distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x E. y  e.  A  ph
 
Theoremnfralya 2379* Not-free for restricted universal quantification where  y and  A are distinct. See nfralxy 2377 for a version with  x and  y distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x A. y  e.  A  ph
 
Theoremnfrexya 2380* Not-free for restricted existential quantification where  y and  A are distinct. See nfrexxy 2378 for a version with  x and  y distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x E. y  e.  A  ph
 
Theoremnfra2xy 2381* Not-free given two restricted quantifiers. (Contributed by Jim Kingdon, 20-Aug-2018.)
 |- 
 F/ y A. x  e.  A  A. y  e.  B  ph
 
Theoremnfre1 2382  x is not free in  E. x  e.  A ph. (Contributed by NM, 19-Mar-1997.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ x E. x  e.  A  ph
 
Theoremr3al 2383* Triple restricted universal quantification. (Contributed by NM, 19-Nov-1995.)
 |-  ( A. x  e.  A  A. y  e.  B  A. z  e.  C  ph  <->  A. x A. y A. z ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  ->  ph )
 )
 
Theoremalral 2384 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
 |-  ( A. x ph  ->  A. x  e.  A  ph )
 
Theoremrexex 2385 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
 |-  ( E. x  e.  A  ph  ->  E. x ph )
 
Theoremrsp 2386 Restricted specialization. (Contributed by NM, 17-Oct-1996.)
 |-  ( A. x  e.  A  ph  ->  ( x  e.  A  ->  ph )
 )
 
Theoremrspe 2387 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
 |-  ( ( x  e.  A  /\  ph )  ->  E. x  e.  A  ph )
 
Theoremrsp2 2388 Restricted specialization. (Contributed by NM, 11-Feb-1997.)
 |-  ( A. x  e.  A  A. y  e.  B  ph  ->  ( ( x  e.  A  /\  y  e.  B )  -> 
 ph ) )
 
Theoremrsp2e 2389 Restricted specialization. (Contributed by FL, 4-Jun-2012.)
 |-  ( ( x  e.  A  /\  y  e.  B  /\  ph )  ->  E. x  e.  A  E. y  e.  B  ph )
 
Theoremrspec 2390 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
 |- 
 A. x  e.  A  ph   =>    |-  ( x  e.  A  -> 
 ph )
 
Theoremrgen 2391 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
 |-  ( x  e.  A  -> 
 ph )   =>    |- 
 A. x  e.  A  ph
 
Theoremrgen2a 2392* Generalization rule for restricted quantification. Note that  x and  y needn't be distinct (and illustrates the use of dvelimor 1910). (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.)
 |-  ( ( x  e.  A  /\  y  e.  A )  ->  ph )   =>    |-  A. x  e.  A  A. y  e.  A  ph
 
Theoremrgenw 2393 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
 |-  ph   =>    |- 
 A. x  e.  A  ph
 
Theoremrgen2w 2394 Generalization rule for restricted quantification. Note that  x and  y needn't be distinct. (Contributed by NM, 18-Jun-2014.)
 |-  ph   =>    |- 
 A. x  e.  A  A. y  e.  B  ph
 
Theoremmprg 2395 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
 |-  ( A. x  e.  A  ph  ->  ps )   &    |-  ( x  e.  A  ->  ph )   =>    |- 
 ps
 
Theoremmprgbir 2396 Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.)
 |-  ( ph  <->  A. x  e.  A  ps )   &    |-  ( x  e.  A  ->  ps )   =>    |-  ph
 
Theoremralim 2397 Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)
 |-  ( A. x  e.  A  ( ph  ->  ps )  ->  ( A. x  e.  A  ph  ->  A. x  e.  A  ps ) )
 
Theoremralimi2 2398 Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.)
 |-  ( ( x  e.  A  ->  ph )  ->  ( x  e.  B  ->  ps ) )   =>    |-  ( A. x  e.  A  ph  ->  A. x  e.  B  ps )
 
Theoremralimia 2399 Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.)
 |-  ( x  e.  A  ->  ( ph  ->  ps )
 )   =>    |-  ( A. x  e.  A  ph  ->  A. x  e.  A  ps )
 
Theoremralimiaa 2400 Inference quantifying both antecedent and consequent. (Contributed by NM, 4-Aug-2007.)
 |-  ( ( x  e.  A  /\  ph )  ->  ps )   =>    |-  ( A. x  e.  A  ph  ->  A. x  e.  A  ps )
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