Theorem rexeqi 2629

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 2625	. 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 1331   E.wrex 2415

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420

This theorem is referenced by:  rexrab2  2846  rexprg  3570  rextpg  3572  rexxp  4678  rexrnmpo  5879  0ct  6985  arch  8967  infssuzex  11631  gcdsupex  11635  gcdsupcl  11636  txbas  12416