Theorem rexeqi 2754

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 2750	. 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 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:  rexrab2  2993  rexprg  3760  rextpg  3762  rexxp  4922  rexrnmpo  6197  0ct  7440  nninfwlpoimlemg  7508  arch  9542  infssuzex  10647  zproddc  12327  gcdsupex  12715  gcdsupcl  12716  dvdsprmpweqnn  13096  4sqlem12  13162  txbas  15285  plyun0  15763