Theorem rexeqdv 2679

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 2673	. 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 1353   E.wrex 2456

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461

This theorem is referenced by:  rexeqbidv  2685  rexeqbidva  2687  fnunirn  5767  cbvexfo  5786  fival  6968  nninfwlpoimlemg  7172  nninfwlpoimlemginf  7173  nninfwlpoim  7175  genipv  7507  exfzdc  10237  zproddc  11582  infssuzex  11944  nninfdcex  11948  ennnfonelemrnh  12411  grppropd  12847  dvdsrpropdg  13269  cnpfval  13588