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Theorem spcegv 2707
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 2228 . 2  |-  F/_ x A
2 nfv 1466 . 2  |-  F/ x ps
3 spcgv.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3spcegf 2702 1  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621
This theorem is referenced by:  spcev  2713  eqeu  2783  absneu  3509  elunii  3653  axpweq  3998  euotd  4072  brcogw  4593  opeldmg  4629  breldmg  4630  dmsnopg  4889  dff3im  5428  elunirn  5527  unielxp  5926  op1steq  5931  tfr0dm  6069  tfrlemibxssdm  6074  tfrlemiex  6078  tfr1onlembxssdm  6090  tfr1onlemex  6094  tfrcllembxssdm  6103  tfrcllemex  6107  frecabcl  6146  ertr  6287  f1oen3g  6451  f1dom2g  6453  f1domg  6455  dom3d  6471  en1  6496  phpelm  6562  isinfinf  6593  ordiso  6708  djudom  6766  recexnq  6928  ltexprlemrl  7148  ltexprlemru  7150  recexprlemm  7162  recexprlemloc  7169  recexprlem1ssl  7171  recexprlem1ssu  7172  frecuzrdgtcl  9784  frecuzrdgfunlem  9791  fihasheqf1oi  10161  zfz1isolem1  10210  climeu  10648  fisum  10742  bj-2inf  11479
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