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Theorem spcegv 2826
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 2319 . 2  |-  F/_ x A
2 nfv 1528 . 2  |-  F/ x ps
3 spcgv.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3spcegf 2821 1  |-  ( A  e.  V  ->  ( ps  ->  E. x ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1353   E.wex 1492    e. wcel 2148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740
This theorem is referenced by:  spcedv  2827  spcev  2833  elabd  2883  eqeu  2908  absneu  3665  elunii  3815  axpweq  4172  euotd  4255  brcogw  4797  opeldmg  4833  breldmg  4834  dmsnopg  5101  dff3im  5662  elunirn  5767  unielxp  6175  op1steq  6180  tfr0dm  6323  tfrlemibxssdm  6328  tfrlemiex  6332  tfr1onlembxssdm  6344  tfr1onlemex  6348  tfrcllembxssdm  6357  tfrcllemex  6361  frecabcl  6400  ertr  6550  f1oen3g  6754  f1dom2g  6756  f1domg  6758  dom3d  6774  en1  6799  phpelm  6866  isinfinf  6897  ordiso  7035  djudom  7092  difinfsn  7099  ctm  7108  enumct  7114  djudoml  7218  djudomr  7219  cc2lem  7265  recexnq  7389  ltexprlemrl  7609  ltexprlemru  7611  recexprlemm  7623  recexprlemloc  7630  recexprlem1ssl  7632  recexprlem1ssu  7633  axpre-suploclemres  7900  frecuzrdgtcl  10412  frecuzrdgfunlem  10419  fihasheqf1oi  10767  zfz1isolem1  10820  climeu  11304  fsum3  11395  uzwodc  12038  eltg3  13560  uptx  13777  xblm  13920  bj-2inf  14693  subctctexmid  14753
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