Theorem ceqsexgv 2724

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.)

Hypothesis

Ref	Expression
ceqsexgv.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
ceqsexgv	|- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )

Distinct variable groups:   x, A   ps, x

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem ceqsexgv

Step	Hyp	Ref	Expression
1		nfv 1461	. 2 |- F/ x ps
2		ceqsexgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	ceqsexg 2723	1 |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  ceqsrexv  2725  clel3g  2729  elxp4  4828  elxp5  4829  dmfco  5262  fndmdif  5293  fndmin  5295  fmptco  5351  rexrnmpt2  5636  brtpos2  5889  xpsnen  6318  prarloc  6693