Theorem raleqbi1dv 2637

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 2629	. 2 |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	ralbidv 2438	. 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 1332   A.wral 2417

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422

This theorem is referenced by:  frforeq2  4275  weeq2  4287  peano5  4520  isoeq4  5713  exmidomni  7022  pitonn  7680  peano1nnnn  7684  peano2nnnn  7685  peano5nnnn  7724  peano5nni  8747  1nn  8755  peano2nn  8756  dfuzi  9185  istopg  12205  isbasisg  12250  basis2  12254  eltg2  12261  ispsmet  12531  ismet  12552  isxmet  12553  metrest  12714  cncfval  12767  bj-indeq  13298  bj-nntrans  13320