Theorem raleqi 2732

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 2728	. 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 1395   A.wral 2508

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-ral 2513

This theorem is referenced by:  ralrab2  2968  ralprg  3717  raltpg  3719  omsinds  4714  ralxp  4865  ralrnmpo  6119  nnnninfeq2  7296  fzprval  10278  fztpval  10279  infssuzex  10453  seq3f1olemp  10737  zsumdc  11895  zproddc  12090  2prm  12649  xpsfrnel  13377  nninfsellemdc  16376  nninfsellemsuc  16378