Theorem raleqbi1dv 2673

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)

Hypothesis

Ref	Expression
raleqd.1	|- ( A = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
raleqbi1dv	|- ( A = B -> ( A. x e. A ph <-> A. x e. B ps ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem raleqbi1dv

Step	Hyp	Ref	Expression
1		raleq 2665	. 2 |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	ralbidv 2470	. 2 |- ( A = B -> ( A. x e. B ph <-> A. x e. B ps ) )
4	1, 3	bitrd 187	1 |- ( A = B -> ( A. x e. A ph <-> A. x e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   A.wral 2448

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453

This theorem is referenced by:  frforeq2  4330  weeq2  4342  peano5  4582  isoeq4  5783  exmidomni  7118  pitonn  7810  peano1nnnn  7814  peano2nnnn  7815  peano5nnnn  7854  peano5nni  8881  1nn  8889  peano2nn  8890  dfuzi  9322  mhmpropd  12689  issubm  12695  istopg  12791  isbasisg  12836  basis2  12840  eltg2  12847  ispsmet  13117  ismet  13138  isxmet  13139  metrest  13300  cncfval  13353  bj-indeq  13964  bj-nntrans  13986