Theorem rexeqi 2695

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 2691	. 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 1364   E.wrex 2473

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478

This theorem is referenced by:  rexrab2  2927  rexprg  3670  rextpg  3672  rexxp  4806  rexrnmpo  6034  0ct  7166  nninfwlpoimlemg  7234  arch  9237  zproddc  11722  infssuzex  12086  gcdsupex  12094  gcdsupcl  12095  dvdsprmpweqnn  12474  4sqlem12  12540  txbas  14426  plyun0  14882