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Theorem rspc2 2852
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 2319 . . . 4  |-  F/_ x D
2 rspc2.1 . . . 4  |-  F/ x ch
31, 2nfralxy 2515 . . 3  |-  F/ x A. y  e.  D  ch
4 rspc2.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
54ralbidv 2477 . . 3  |-  ( x  =  A  ->  ( A. y  e.  D  ph  <->  A. y  e.  D  ch ) )
63, 5rspc 2835 . 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 2835 . 2  |-  ( B  e.  D  ->  ( A. y  e.  D  ch  ->  ps ) )
106, 9sylan9 409 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    /\ wa 104    <-> wb 105    = wceq 1353   F/wnf 1460    e. wcel 2148   A.wral 2455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739
This theorem is referenced by:  rspc2v  2854  disjiun  3995
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