Theorem raleqi 2630

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 2626	. 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 104   = wceq 1331   A.wral 2416

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 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421

This theorem is referenced by:  ralrab2  2849  ralprg  3574  raltpg  3576  omsinds  4535  ralxp  4682  ralrnmpo  5885  fzprval  9862  fztpval  9863  seq3f1olemp  10275  zsumdc  11153  infssuzex  11642  2prm  11808  nninfsellemdc  13206  nninfsellemsuc  13208