Theorem rexeqdv 2712

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 2706	. 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 1373   E.wrex 2487

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492

This theorem is referenced by:  rexeqtrdv  2714  rexeqtrrdv  2716  rexeqbidv  2722  rexeqbidva  2724  fnunirn  5859  cbvexfo  5878  fival  7098  nninfwlpoimlemg  7303  nninfwlpoimlemginf  7304  nninfwlpoim  7307  nninfinfwlpo  7308  genipv  7657  exfzdc  10406  infssuzex  10413  nninfdcex  10417  zproddc  12005  ennnfonelemrnh  12902  grppropd  13464  dvdsrpropdg  14024  znunit  14536  cnpfval  14782  plyval  15319  uhgrvtxedgiedgb  15847