Theorem rexeqdv 2756

Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)

Hypothesis

Ref	Expression
raleq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
rexeqdv	|- ( ph -> ( E. x e. A ps <-> E. x e. B ps ) )

Distinct variable groups:   x, A   x, B

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

Proof of Theorem rexeqdv

Step	Hyp	Ref	Expression
1		raleq1d.1	. 2 |- ( ph -> A = B )
2		rexeq 2750	. 2 |- ( A = B -> ( E. x e. A ps <-> E. x e. B ps ) )
3	1, 2	syl 14	1 |- ( ph -> ( E. x e. A ps <-> E. x e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   E.wrex 2529

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534

This theorem is referenced by:  rexeqtrdv  2758  rexeqtrrdv  2760  rexeqbidv  2766  rexeqbidva  2768  fnunirn  5963  cbvexfo  5982  fival  7294  nninfwlpoimlemg  7505  nninfwlpoimlemginf  7506  nninfwlpoim  7509  nninfinfwlpo  7510  genipv  7866  exfzdc  10637  infssuzex  10644  nninfdcex  10650  zproddc  12324  ennnfonelemrnh  13285  grppropd  13799  dvdsrpropdg  14427  znunit  14966  cnpfval  15219  plyval  15756  uhgrvtxedgiedgb  16298  wlkvtxedg  16518