Theorem raleqi 2697

Description: Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypothesis

Ref	Expression
raleq1i.1	|- A = B

Assertion

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

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem raleqi

Step	Hyp	Ref	Expression
1		raleq1i.1	. 2 |- A = B
2		raleq 2693	. 2 |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )
3	1, 2	ax-mp 5	1 |- ( A. x e. A ph <-> A. x e. B ph )

Syntax hints:   <-> wb 105   = wceq 1364   A.wral 2475

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480

This theorem is referenced by:  ralrab2  2929  ralprg  3673  raltpg  3675  omsinds  4658  ralxp  4809  ralrnmpo  6037  nnnninfeq2  7195  fzprval  10157  fztpval  10158  infssuzex  10323  seq3f1olemp  10607  zsumdc  11549  zproddc  11744  2prm  12295  xpsfrnel  12987  nninfsellemdc  15654  nninfsellemsuc  15656