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Theorem rspc2 3057
Description: 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.)
Hypotheses
Ref Expression
rspc2.1  |-  F/ x ch
rspc2.2  |-  F/ y ps
rspc2.3  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
rspc2.4  |-  ( y  =  B  ->  ( ch 
<->  ps ) )
Assertion
Ref Expression
rspc2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A. x  e.  C  A. y  e.  D  ph  ->  ps ) )
Distinct variable groups:    x, y, A    y, B    x, C    x, D, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    ch( x, y)    B( x)    C( y)

Proof of Theorem rspc2
StepHypRef Expression
1 nfcv 2572 . . . 4  |-  F/_ x D
2 rspc2.1 . . . 4  |-  F/ x ch
31, 2nfral 2759 . . 3  |-  F/ x A. y  e.  D  ch
4 rspc2.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
54ralbidv 2725 . . 3  |-  ( x  =  A  ->  ( A. y  e.  D  ph  <->  A. y  e.  D  ch ) )
63, 5rspc 3046 . 2  |-  ( A  e.  C  ->  ( A. x  e.  C  A. y  e.  D  ph 
->  A. y  e.  D  ch ) )
7 rspc2.2 . . 3  |-  F/ y ps
8 rspc2.4 . . 3  |-  ( y  =  B  ->  ( ch 
<->  ps ) )
97, 8rspc 3046 . 2  |-  ( B  e.  D  ->  ( A. y  e.  D  ch  ->  ps ) )
106, 9sylan9 639 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A. x  e.  C  A. y  e.  D  ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1553    = wceq 1652    e. wcel 1725   A.wral 2705
This theorem is referenced by:  rspc2v  3058  dvmptfsum  19859  fphpd  26877
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-v 2958
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