Theorem rexeqi 2670

Description: Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)

Hypothesis

Ref	Expression
raleq1i.1	|- A = B

Assertion

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

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem rexeqi

Step	Hyp	Ref	Expression
1		raleq1i.1	. 2 |- A = B
2		rexeq 2666	. 2 |- ( A = B -> ( E. x e. A ph <-> E. x e. B ph ) )
3	1, 2	ax-mp 5	1 |- ( E. x e. A ph <-> E. x e. B ph )

Syntax hints:   <-> wb 104   = wceq 1348   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454

This theorem is referenced by:  rexrab2  2897  rexprg  3635  rextpg  3637  rexxp  4755  rexrnmpo  5968  0ct  7084  nninfwlpoimlemg  7151  arch  9132  zproddc  11542  infssuzex  11904  gcdsupex  11912  gcdsupcl  11913  dvdsprmpweqnn  12289  txbas  13052