Theorem raleqi 2706

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 2702	. 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 1373   A.wral 2484

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489

This theorem is referenced by:  ralrab2  2938  ralprg  3684  raltpg  3686  omsinds  4671  ralxp  4822  ralrnmpo  6062  nnnninfeq2  7233  fzprval  10206  fztpval  10207  infssuzex  10378  seq3f1olemp  10662  zsumdc  11728  zproddc  11923  2prm  12482  xpsfrnel  13209  nninfsellemdc  15984  nninfsellemsuc  15986