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Theorem List for Intuitionistic Logic Explorer - 2401-2500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-rmo 2401 Define restricted "at most one". (Contributed by NM, 16-Jun-2017.)
 |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
 )
 
Definitiondf-rab 2402 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 2403 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
 |-  ( A. x  e.  A  -.  ph  <->  -.  E. x  e.  A  ph )
 
Theoremrexnalim 2404 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 2405 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 2406 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 2407 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 2408 Formula-building rule for restricted universal quantifier (deduction form). (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 2409 Formula-building rule for restricted existential quantifier (deduction form). (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 2410* Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 4-Mar-1997.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  A  ch ) )
 
Theoremrexbidva 2411* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 9-Mar-1997.)
 |-  ( ( ph  /\  x  e.  A )  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  A  ch ) )
 
Theoremralbid 2412 Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  A  ch ) )
 
Theoremrexbid 2413 Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 27-Jun-1998.)
 |- 
 F/ x ph   &    |-  ( ph  ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  A  ch ) )
 
Theoremralbidv 2414* Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 20-Nov-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  A  ch ) )
 
Theoremrexbidv 2415* Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 20-Nov-1994.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  A  ch ) )
 
Theoremralbidv2 2416* Formula-building rule for restricted universal quantifier (deduction form). (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 2417* Formula-building rule for restricted existential quantifier (deduction form). (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 2418 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 2419 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 2420 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 2421 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 2422 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 2423 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 2424 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 2425 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 2426 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 2427 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 2428* 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 2429* 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 2430* 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 2431* 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 2432* 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 2433* Formula-building rule for restricted universal quantifier (deduction form). (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 2434* Formula-building rule for restricted universal quantifiers (deduction form). (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 2435* Formula-building rule for restricted existential quantifiers (deduction form). (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 2436* Formula-building rule for restricted universal quantifiers (deduction form). (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 2437* Formula-building rule for restricted existential quantifiers (deduction form). (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 2438* Formula-building rule for restricted quantifiers (deduction form). (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 2439 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 2440* Two ways to say " A belongs to  B." (Contributed by NM, 22-Nov-1994.)
 |-  ( A  e.  B  <->  E. x  e.  B  x  =  A )
 
Theoremhbral 2441 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 2442  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 2443  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 2444* Not-free for restricted universal quantification where  x and  y are distinct. See nfraldya 2446 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 2445* Not-free for restricted existential quantification where  x and  y are distinct. See nfrexdya 2447 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 2446* Not-free for restricted universal quantification where  y and  A are distinct. See nfraldxy 2444 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 2447* Not-free for restricted existential quantification where  y and  A are distinct. See nfrexdxy 2445 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 2448* Not-free for restricted universal quantification where  x and  y are distinct. See nfralya 2450 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 2449* Not-free for restricted existential quantification where  x and  y are distinct. See nfrexya 2451 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 2450* Not-free for restricted universal quantification where  y and  A are distinct. See nfralxy 2448 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 2451* Not-free for restricted existential quantification where  y and  A are distinct. See nfrexxy 2449 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 2452* 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 2453  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 2454* 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 2455 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
 |-  ( A. x ph  ->  A. x  e.  A  ph )
 
Theoremrexex 2456 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
 |-  ( E. x  e.  A  ph  ->  E. x ph )
 
Theoremrsp 2457 Restricted specialization. (Contributed by NM, 17-Oct-1996.)
 |-  ( A. x  e.  A  ph  ->  ( x  e.  A  ->  ph )
 )
 
Theoremrspe 2458 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
 |-  ( ( x  e.  A  /\  ph )  ->  E. x  e.  A  ph )
 
Theoremrsp2 2459 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 2460 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 2461 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
 |- 
 A. x  e.  A  ph   =>    |-  ( x  e.  A  -> 
 ph )
 
Theoremrgen 2462 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
 |-  ( x  e.  A  -> 
 ph )   =>    |- 
 A. x  e.  A  ph
 
Theoremrgen2a 2463* Generalization rule for restricted quantification. Note that  x and  y needn't be distinct (and illustrates the use of dvelimor 1971). (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 2464 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
 |-  ph   =>    |- 
 A. x  e.  A  ph
 
Theoremrgen2w 2465 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 2466 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
 |-  ( A. x  e.  A  ph  ->  ps )   &    |-  ( x  e.  A  ->  ph )   =>    |- 
 ps
 
Theoremmprgbir 2467 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 2468 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 2469 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 2470 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 2471 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 2472 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 )
 
Theorem2ralimi 2473 Inference quantifying both antecedent and consequent two times, with strong hypothesis. (Contributed by AV, 3-Dec-2021.)
 |-  ( ph  ->  ps )   =>    |-  ( A. x  e.  A  A. y  e.  B  ph  ->  A. x  e.  A  A. y  e.  B  ps )
 
Theoremral2imi 2474 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 2475 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 2476* 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 2477* 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 ) )
 
Theoremralimdvva 2478* Deduction doubly quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90 (alim 1418). (Contributed by AV, 27-Nov-2019.)
 |-  ( ( 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 ) )
 
Theoremralimdv2 2479* 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 2480 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 2481* 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 2482* 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 2483* 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 2484 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 2485 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 2486* 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 2487 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 2488* 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 2489* 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 2490* 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 )
 
Theoremralrimivva 2491* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  ps )   =>    |-  ( ph  ->  A. x  e.  A  A. y  e.  B  ps )
 
Theoremralrimivvva 2492* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with triple quantification.) (Contributed by Mario Carneiro, 9-Jul-2014.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B  /\  z  e.  C )
 )  ->  ps )   =>    |-  ( ph  ->  A. x  e.  A  A. y  e.  B  A. z  e.  C  ps )
 
Theoremralrimdvv 2493* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 1-Jun-2005.)
 |-  ( ph  ->  ( ps  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ch )
 ) )   =>    |-  ( ph  ->  ( ps  ->  A. x  e.  A  A. y  e.  B  ch ) )
 
Theoremralrimdvva 2494* Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version with double quantification.) (Contributed by NM, 2-Feb-2008.)
 |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B )
 )  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x  e.  A  A. y  e.  B  ch ) )
 
Theoremrgen2 2495* Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.)
 |-  ( ( x  e.  A  /\  y  e.  B )  ->  ph )   =>    |-  A. x  e.  A  A. y  e.  B  ph
 
Theoremrgen3 2496* Generalization rule for restricted quantification. (Contributed by NM, 12-Jan-2008.)
 |-  ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  ->  ph )   =>    |-  A. x  e.  A  A. y  e.  B  A. z  e.  C  ph
 
Theoremr19.21bi 2497 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)
 |-  ( ph  ->  A. x  e.  A  ps )   =>    |-  ( ( ph  /\  x  e.  A ) 
 ->  ps )
 
Theoremrspec2 2498 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
 |- 
 A. x  e.  A  A. y  e.  B  ph   =>    |-  (
 ( x  e.  A  /\  y  e.  B )  ->  ph )
 
Theoremrspec3 2499 Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.)
 |- 
 A. x  e.  A  A. y  e.  B  A. z  e.  C  ph   =>    |-  ( ( x  e.  A  /\  y  e.  B  /\  z  e.  C )  ->  ph )
 
Theoremr19.21be 2500 Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.)
 |-  ( ph  ->  A. x  e.  A  ps )   =>    |-  A. x  e.  A  ( ph  ->  ps )
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