Theorem rexeqi 2733

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 2729	. 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 1395   E.wrex 2509

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514

This theorem is referenced by:  rexrab2  2970  rexprg  3718  rextpg  3720  rexxp  4866  rexrnmpo  6120  0ct  7274  nninfwlpoimlemg  7342  arch  9366  infssuzex  10453  zproddc  12090  gcdsupex  12478  gcdsupcl  12479  dvdsprmpweqnn  12859  4sqlem12  12925  txbas  14932  plyun0  15410