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Theorem rspc2 3215
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 2616 . . . 4  |-  F/_ x D
2 rspc2.1 . . . 4  |-  F/ x ch
31, 2nfral 2840 . . 3  |-  F/ x A. y  e.  D  ch
4 rspc2.3 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
54ralbidv 2893 . . 3  |-  ( x  =  A  ->  ( A. y  e.  D  ph  <->  A. y  e.  D  ch ) )
63, 5rspc 3201 . 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 3201 . 2  |-  ( B  e.  D  ->  ( A. y  e.  D  ch  ->  ps ) )
106, 9sylan9 655 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A. x  e.  C  A. y  e.  D  ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   F/wnf 1621    e. wcel 1823   A.wral 2804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-v 3108
This theorem is referenced by:  rspc2v  3216  reu2eqd  3293  fvmpt2curryd  6992  dvmptfsum  22542  fphpd  30989
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