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Theorem spcegv 2746
Description: Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
Hypothesis
Ref Expression
spcgv.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
spcegv  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem spcegv
StepHypRef Expression
1 nfcv 2256 . 2  |-  F/_ x A
2 nfv 1491 . 2  |-  F/ x ps
3 spcgv.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3spcegf 2741 1  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463
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:  spcev  2752  eqeu  2825  absneu  3563  elunii  3709  axpweq  4063  euotd  4144  brcogw  4676  opeldmg  4712  breldmg  4713  dmsnopg  4978  dff3im  5531  elunirn  5633  unielxp  6038  op1steq  6043  tfr0dm  6185  tfrlemibxssdm  6190  tfrlemiex  6194  tfr1onlembxssdm  6206  tfr1onlemex  6210  tfrcllembxssdm  6219  tfrcllemex  6223  frecabcl  6262  ertr  6410  f1oen3g  6614  f1dom2g  6616  f1domg  6618  dom3d  6634  en1  6659  phpelm  6726  isinfinf  6757  ordiso  6887  djudom  6944  difinfsn  6951  ctm  6960  enumct  6966  djudoml  7039  djudomr  7040  recexnq  7162  ltexprlemrl  7382  ltexprlemru  7384  recexprlemm  7396  recexprlemloc  7403  recexprlem1ssl  7405  recexprlem1ssu  7406  axpre-suploclemres  7673  frecuzrdgtcl  10136  frecuzrdgfunlem  10143  fihasheqf1oi  10485  zfz1isolem1  10534  climeu  11016  fsum3  11107  eltg3  12132  uptx  12349  xblm  12492  bj-2inf  12970  subctctexmid  13030
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