Theorem raleqi 2566

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 2562	. 2 |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )
3	1, 2	ax-mp 7	1 |- ( A. x e. A ph <-> A. x e. B ph )

Syntax hints:   <-> wb 103   = wceq 1289   A.wral 2359

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364

This theorem is referenced by:  ralrab2  2780  ralprg  3493  raltpg  3495  omsinds  4435  ralxp  4579  ralrnmpt2  5759  fzprval  9496  fztpval  9497  seq3f1olemp  9931  zisum  10774  infssuzex  11223  2prm  11387  nninfsellemdc  11902  nninfsellemsuc  11904