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Theorem rspc2 2889
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 2419 . . . 4  |-  F/_ x D
2 rspc2.1 . . . 4  |-  F/ x ch
31, 2nfral 2596 . . 3  |-  F/ x A. y  e.  D  ch
4 rspc2.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
54ralbidv 2563 . . 3  |-  ( x  =  A  ->  ( A. y  e.  D  ph  <->  A. y  e.  D  ch ) )
63, 5rspc 2878 . 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 2878 . 2  |-  ( B  e.  D  ->  ( A. y  e.  D  ch  ->  ps ) )
106, 9sylan9 638 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 176    /\ wa 358   F/wnf 1531    = wceq 1623    e. wcel 1684   A.wral 2543
This theorem is referenced by:  rspc2v  2890  dvmptfsum  19322  fphpd  26899
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790
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