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Theorem List for Intuitionistic Logic Explorer - 2301-2400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremneleq1 2301 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
 |-  ( A  =  B  ->  ( A  e/  C  <->  B 
 e/  C ) )
 
Theoremneleq2 2302 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
 |-  ( A  =  B  ->  ( C  e/  A  <->  C 
 e/  B ) )
 
Theoremneleq12d 2303 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 2304 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 2305 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 2306 Extend wff notation to include restricted universal quantification.
 wff  A. x  e.  A  ph
 
Syntaxwrex 2307 Extend wff notation to include restricted existential quantification.
 wff  E. x  e.  A  ph
 
Syntaxwreu 2308 Extend wff notation to include restricted existential uniqueness.
 wff  E! x  e.  A  ph
 
Syntaxwrmo 2309 Extend wff notation to include restricted "at most one."
 wff  E* x  e.  A  ph
 
Syntaxcrab 2310 Extend class notation to include the restricted class abstraction (class builder).
 class  { x  e.  A  |  ph }
 
Definitiondf-ral 2311 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 2312 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 2313 Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
 |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
 )
 
Definitiondf-rmo 2314 Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
 |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
 )
 
Definitiondf-rab 2315 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 2316 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
 |-  ( A. x  e.  A  -.  ph  <->  -.  E. x  e.  A  ph )
 
Theoremrexnalim 2317 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 2318 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 2319 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 2320 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 2321 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 2322* 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 2323* 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 2324 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 2325 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 2326* 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 2327* 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 2328* 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 2329* 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 2330 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 2331 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 2332 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 2333 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 2334 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 2335 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 2336 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 2337 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 2338 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 2339 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 2340* 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 2341* 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 2342* 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 2343* 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 2344* 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 2345* 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 2346* 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 2347* 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 2348* 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 2349* 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 2350* 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 2351 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 2352* Two ways to say " A belongs to  B." (Contributed by NM, 22-Nov-1994.)
 |-  ( A  e.  B  <->  E. x  e.  B  x  =  A )
 
Theoremhbral 2353 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 2354  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 2355  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 2356* Not-free for restricted universal quantification where  x and  y are distinct. See nfraldya 2358 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 2357* Not-free for restricted existential quantification where  x and  y are distinct. See nfrexdya 2359 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 2358* Not-free for restricted universal quantification where  y and  A are distinct. See nfraldxy 2356 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 2359* Not-free for restricted existential quantification where  y and  A are distinct. See nfrexdxy 2357 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 2360* Not-free for restricted universal quantification where  x and  y are distinct. See nfralya 2362 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 2361* Not-free for restricted existential quantification where  x and  y are distinct. See nfrexya 2363 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 2362* Not-free for restricted universal quantification where  y and  A are distinct. See nfralxy 2360 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 2363* Not-free for restricted existential quantification where  y and  A are distinct. See nfrexxy 2361 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 2364* 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 2365  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 2366* 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 2367 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
 |-  ( A. x ph  ->  A. x  e.  A  ph )
 
Theoremrexex 2368 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
 |-  ( E. x  e.  A  ph  ->  E. x ph )
 
Theoremrsp 2369 Restricted specialization. (Contributed by NM, 17-Oct-1996.)
 |-  ( A. x  e.  A  ph  ->  ( x  e.  A  ->  ph )
 )
 
Theoremrspe 2370 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
 |-  ( ( x  e.  A  /\  ph )  ->  E. x  e.  A  ph )
 
Theoremrsp2 2371 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 2372 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 2373 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
 |- 
 A. x  e.  A  ph   =>    |-  ( x  e.  A  -> 
 ph )
 
Theoremrgen 2374 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
 |-  ( x  e.  A  -> 
 ph )   =>    |- 
 A. x  e.  A  ph
 
Theoremrgen2a 2375* Generalization rule for restricted quantification. Note that  x and  y needn't be distinct (and illustrates the use of dvelimor 1894). (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 2376 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
 |-  ph   =>    |- 
 A. x  e.  A  ph
 
Theoremrgen2w 2377 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 2378 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
 |-  ( A. x  e.  A  ph  ->  ps )   &    |-  ( x  e.  A  ->  ph )   =>    |- 
 ps
 
Theoremmprgbir 2379 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 2380 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 2381 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 2382 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 2383 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 )
 
Theoremralimi 2384 Inference quantifying both antecedent and consequent, with strong hypothesis. (Contributed by NM, 4-Mar-1997.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x  e.  A  ph 
 ->  A. x  e.  A  ps )
 
Theoremral2imi 2385 Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x  e.  A  ph  ->  ( A. x  e.  A  ps  ->  A. x  e.  A  ch ) )
 
Theoremralimdaa 2386 Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-Sep-2003.)
 |- 
 F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  ->  ch )
 )   =>    |-  ( ph  ->  ( A. x  e.  A  ps  ->  A. x  e.  A  ch ) )
 
Theoremralimdva 2387* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 22-May-1999.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps  ->  A. x  e.  A  ch ) )
 
Theoremralimdv 2388* Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps  ->  A. x  e.  A  ch ) )
 
Theoremralimdv2 2389* Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.)
 |-  ( ph  ->  (
 ( x  e.  A  ->  ps )  ->  ( x  e.  B  ->  ch ) ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps  ->  A. x  e.  B  ch ) )
 
Theoremralrimi 2390 Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( x  e.  A  ->  ps ) )   =>    |-  ( ph  ->  A. x  e.  A  ps )
 
Theoremralrimiv 2391* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ph  ->  ( x  e.  A  ->  ps ) )   =>    |-  ( ph  ->  A. x  e.  A  ps )
 
Theoremralrimiva 2392* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Jan-2006.)
 |-  ( ( ph  /\  x  e.  A )  ->  ps )   =>    |-  ( ph  ->  A. x  e.  A  ps )
 
Theoremralrimivw 2393* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x  e.  A  ps )
 
Theoremr19.21t 2394 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)
 |-  ( F/ x ph  ->  ( A. x  e.  A  ( ph  ->  ps )  <->  ( ph  ->  A. x  e.  A  ps ) ) )
 
Theoremr19.21 2395 Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by Scott Fenton, 30-Mar-2011.)
 |- 
 F/ x ph   =>    |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( ph  ->  A. x  e.  A  ps ) )
 
Theoremr19.21v 2396* Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)
 |-  ( A. x  e.  A  ( ph  ->  ps )  <->  ( ph  ->  A. x  e.  A  ps ) )
 
Theoremralrimd 2397 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)
 |- 
 F/ x ph   &    |-  F/ x ps   &    |-  ( ph  ->  ( ps  ->  ( x  e.  A  ->  ch ) ) )   =>    |-  ( ph  ->  ( ps  ->  A. x  e.  A  ch ) )
 
Theoremralrimdv 2398* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998.)
 |-  ( ph  ->  ( ps  ->  ( x  e.  A  ->  ch )
 ) )   =>    |-  ( ph  ->  ( ps  ->  A. x  e.  A  ch ) )
 
Theoremralrimdva 2399* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 2-Feb-2008.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x  e.  A  ch ) )
 
Theoremralrimivv 2400* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 24-Jul-2004.)
 |-  ( ph  ->  (
 ( x  e.  A  /\  y  e.  B )  ->  ps ) )   =>    |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ps )
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