Theorem ceqsexgv 2946

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 1577	. 2 |- F/ x ps
2		ceqsexgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	ceqsexg 2945	1 |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2203

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815

This theorem is referenced by:  ceqsrexv  2947  clel3g  2951  elxp4  5250  elxp5  5251  dmfco  5745  fndmdif  5783  fndmin  5785  fmptco  5843  rexrnmpo  6169  brtpos2  6482  xpsnen  7072  prarloc  7818  pceu  12993  4sqlem12  13100  znleval  14801  metrest  15371