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Theorem spcegf 2741
Description: Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
Hypotheses
Ref Expression
spcgf.1  |-  F/_ x A
spcgf.2  |-  F/ x ps
spcgf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
spcegf  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)

Proof of Theorem spcegf
StepHypRef Expression
1 spcgf.2 . . 3  |-  F/ x ps
2 spcgf.1 . . 3  |-  F/_ x A
31, 2spcegft 2737 . 2  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  ( A  e.  V  ->  ( ps  ->  E. x ph ) ) )
4 spcgf.3 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
53, 4mpg 1410 1  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1314   F/wnf 1419   E.wex 1451    e. wcel 1463   F/_wnfc 2243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660
This theorem is referenced by:  spcegv  2746  rspce  2756  euotd  4144  seq3f1olemstep  10225
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