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Theorem List for Metamath Proof Explorer - 2501-2600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnelne1 2501 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 2502 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 )
 
Theoremneleq1 2503 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
 |-  ( A  =  B  ->  ( A  e/  C  <->  B 
 e/  C ) )
 
Theoremneleq2 2504 Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)
 |-  ( A  =  B  ->  ( C  e/  A  <->  C 
 e/  B ) )
 
Theoremnfne 2505 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
 
Theoremnfnel 2506 Bound-variable hypothesis builder for inequality. (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
 
Theoremnfned 2507 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 )
 
Theoremnfneld 2508 Bound-variable hypothesis builder for inequality. (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 2509 Extend wff notation to include restricted universal quantification.
 wff  A. x  e.  A  ph
 
Syntaxwrex 2510 Extend wff notation to include restricted existential quantification.
 wff  E. x  e.  A  ph
 
Syntaxwreu 2511 Extend wff notation to include restricted existential uniqueness.
 wff  E! x  e.  A  ph
 
Syntaxcrab 2512 Extend class notation to include the restricted class abstraction (class builder).
 class  { x  e.  A  |  ph }
 
Definitiondf-ral 2513 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 2514 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 2515 Define restricted existential uniqueness. (Contributed by NM, 22-Nov-1994.)
 |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
 )
 
Definitiondf-rab 2516 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 2517 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
 |-  ( A. x  e.  A  -.  ph  <->  -.  E. x  e.  A  ph )
 
Theoremrexnal 2518 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
 |-  ( E. x  e.  A  -.  ph  <->  -.  A. x  e.  A  ph )
 
Theoremdfral2 2519 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
 |-  ( A. x  e.  A  ph  <->  -.  E. x  e.  A  -.  ph )
 
Theoremdfrex2 2520 Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)
 |-  ( E. x  e.  A  ph  <->  -.  A. x  e.  A  -.  ph )
 
Theoremralbida 2521 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 2522 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 2523* 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 2524* 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 2525 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 2526 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 2527* 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 2528* 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 2529* 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 2530* 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 2531 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 2532 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 2533 Inference adding 2 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 2534 Inference adding 2 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 2535 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 2536 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 )
 
Theoremralbiia 2537 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 2538 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 2539* Inference adding 2 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 2540* 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 2541* 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 2542* 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 2543* 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 2544* 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 2545* 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 2546* 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 2547* 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 2548* 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 2549* 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 2550 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 ) )
 
Theoremrexanali 2551 A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)
 |-  ( E. x  e.  A  ( ph  /\  -.  ps )  <->  -.  A. x  e.  A  ( ph  ->  ps ) )
 
Theoremrisset 2552* Two ways to say " A belongs to  B." (Contributed by NM, 22-Nov-1994.)
 |-  ( A  e.  B  <->  E. x  e.  B  x  =  A )
 
Theoremhbral 2553 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 2554  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 2555  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
 
Theoremnfrald 2556 Deduction version of nfral 2558. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y  e.  A  ps )
 
Theoremnfrexd 2557 Deduction version of nfrex 2560. (Contributed by Mario Carneiro, 14-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E. y  e.  A  ps )
 
Theoremnfral 2558 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x A. y  e.  A  ph
 
Theoremnfra2 2559* Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 27423. Contributed by Alan Sare 31-Dec-2011. (Contributed by NM, 31-Dec-2011.)
 |- 
 F/ y A. x  e.  A  A. y  e.  B  ph
 
Theoremnfrex 2560 Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |-  F/_ x A   &    |-  F/ x ph   =>    |-  F/ x E. y  e.  A  ph
 
Theoremnfre1 2561  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 2562* 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 2563 Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.)
 |-  ( A. x ph  ->  A. x  e.  A  ph )
 
Theoremrexex 2564 Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
 |-  ( E. x  e.  A  ph  ->  E. x ph )
 
Theoremra4 2565 Restricted specialization. (Contributed by NM, 17-Oct-1996.)
 |-  ( A. x  e.  A  ph  ->  ( x  e.  A  ->  ph )
 )
 
Theoremra4e 2566 Restricted specialization. (Contributed by NM, 12-Oct-1999.)
 |-  ( ( x  e.  A  /\  ph )  ->  E. x  e.  A  ph )
 
Theoremra42 2567 Restricted specialization. (Contributed by NM, 11-Feb-1997.)
 |-  ( A. x  e.  A  A. y  e.  B  ph  ->  ( ( x  e.  A  /\  y  e.  B )  -> 
 ph ) )
 
Theoremra42e 2568 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 2569 Specialization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
 |- 
 A. x  e.  A  ph   =>    |-  ( x  e.  A  -> 
 ph )
 
Theoremrgen 2570 Generalization rule for restricted quantification. (Contributed by NM, 19-Nov-1994.)
 |-  ( x  e.  A  -> 
 ph )   =>    |- 
 A. x  e.  A  ph
 
Theoremrgen2a 2571* Generalization rule for restricted quantification. Note that  x and  y needn't be distinct (and illustrates the use of dvelim 2092). (Contributed by NM, 23-Nov-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.
 |-  ( ( x  e.  A  /\  y  e.  A )  ->  ph )   =>    |-  A. x  e.  A  A. y  e.  A  ph
 
Theoremrgenw 2572 Generalization rule for restricted quantification. (Contributed by NM, 18-Jun-2014.)
 |-  ph   =>    |- 
 A. x  e.  A  ph
 
Theoremrgen2w 2573 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 2574 Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.)
 |-  ( A. x  e.  A  ph  ->  ps )   &    |-  ( x  e.  A  ->  ph )   =>    |- 
 ps
 
Theoremmprgbir 2575 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 2576 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 2577 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 2578 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 2579 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 2580 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 2581 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 2582 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 2583* 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 2584* 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 2585* 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 2586 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 2587* 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 2588* 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 2589* 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 2590 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 2591 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 2592* 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 2593 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 2594* 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 2595* 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 2596* 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 2597* 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 2598* 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 2599* 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 2600* 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 ) )
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