Theorem rexeqdv 2700

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 2694	. 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 1364   E.wrex 2476

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481

This theorem is referenced by:  rexeqtrdv  2702  rexeqtrrdv  2704  rexeqbidv  2710  rexeqbidva  2712  fnunirn  5817  cbvexfo  5836  fival  7045  nninfwlpoimlemg  7250  nninfwlpoimlemginf  7251  nninfwlpoim  7253  genipv  7593  exfzdc  10333  infssuzex  10340  nninfdcex  10344  zproddc  11761  ennnfonelemrnh  12658  grppropd  13219  dvdsrpropdg  13779  znunit  14291  cnpfval  14515  plyval  15052