Theorem raleqbi1dv 2513

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 2505	. 2 |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	ralbidv 2326	. 2 |- ( A = B -> ( A. x e. B ph <-> A. x e. B ps ) )
4	1, 3	bitrd 177	1 |- ( A = B -> ( A. x e. A ph <-> A. x e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311

This theorem is referenced by:  frforeq2  4082  weeq2  4094  peano5  4321  isoeq4  5444  pitonn  6924  peano1nnnn  6928  peano2nnnn  6929  peano5nnnn  6966  peano5nni  7917  1nn  7925  peano2nn  7926  dfuzi  8348  bj-indeq  10053  bj-nntrans  10076