Theorem spcegv 2641

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

Step	Hyp	Ref	Expression
1		nfcv 2178	. 2 |- F/_ x A
2		nfv 1421	. 2 |- F/ x ps
3		spcgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
4	1, 2, 3	spcegf 2636	1 |- ( A e. V -> ( ps -> E. x ph ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559

This theorem is referenced by:  spcev  2647  eqeu  2711  absneu  3442  elunii  3585  axpweq  3924  euotd  3991  brcogw  4504  opeldmg  4540  breldmg  4541  dmsnopg  4792  dff3im  5312  elunirn  5405  unielxp  5800  op1steq  5805  tfr0  5937  tfrlemibxssdm  5941  tfrlemiex  5945  ertr  6121  f1oen3g  6234  f1dom2g  6236  f1domg  6238  dom3d  6254  en1  6279  phpelm  6328  ordiso  6358  recexnq  6488  ltexprlemrl  6708  ltexprlemru  6710  recexprlemm  6722  recexprlemloc  6729  recexprlem1ssl  6731  recexprlem1ssu  6732  frecuzrdgfn  9198  climeu  9817  bj-2inf  10062