Theorem raleqbi1dv 2669

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 2661	. 2 |- ( A = B -> ( A. x e. A ph <-> A. x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	ralbidv 2466	. 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 1343   A.wral 2444

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449

This theorem is referenced by:  frforeq2  4323  weeq2  4335  peano5  4575  isoeq4  5772  exmidomni  7106  pitonn  7789  peano1nnnn  7793  peano2nnnn  7794  peano5nnnn  7833  peano5nni  8860  1nn  8868  peano2nn  8869  dfuzi  9301  istopg  12637  isbasisg  12682  basis2  12686  eltg2  12693  ispsmet  12963  ismet  12984  isxmet  12985  metrest  13146  cncfval  13199  bj-indeq  13811  bj-nntrans  13833