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Theorem rspc2 2683
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 2194 . . . 4  |-  F/_ x D
2 rspc2.1 . . . 4  |-  F/ x ch
31, 2nfralxy 2377 . . 3  |-  F/ x A. y  e.  D  ch
4 rspc2.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
54ralbidv 2343 . . 3  |-  ( x  =  A  ->  ( A. y  e.  D  ph  <->  A. y  e.  D  ch ) )
63, 5rspc 2667 . 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 2667 . 2  |-  ( B  e.  D  ->  ( A. y  e.  D  ch  ->  ps ) )
106, 9sylan9 395 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 101    <-> wb 102    = wceq 1259   F/wnf 1365    e. wcel 1409   A.wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576
This theorem is referenced by:  rspc2v  2685
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