Theorem ceqsexgv 2889

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 1539	. 2 |- F/ x ps
2		ceqsexgv.1	. 2 |- ( x = A -> ( ph <-> ps ) )
3	1, 2	ceqsexg 2888	1 |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2164

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762

This theorem is referenced by:  ceqsrexv  2890  clel3g  2894  elxp4  5153  elxp5  5154  dmfco  5625  fndmdif  5663  fndmin  5665  fmptco  5724  rexrnmpo  6034  brtpos2  6304  xpsnen  6875  prarloc  7563  pceu  12433  4sqlem12  12540  znleval  14141  metrest  14674