ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  raleqbi1dv Unicode version

Theorem raleqbi1dv 2570
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
StepHypRef Expression
1 raleq 2562 . 2  |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ph ) )
2 raleqd.1 . . 3  |-  ( A  =  B  ->  ( ph 
<->  ps ) )
32ralbidv 2380 . 2  |-  ( A  =  B  ->  ( A. x  e.  B  ph  <->  A. x  e.  B  ps ) )
41, 3bitrd 186 1  |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289   A.wral 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364
This theorem is referenced by:  frforeq2  4163  weeq2  4175  peano5  4403  isoeq4  5565  exmidomni  6777  pitonn  7364  peano1nnnn  7368  peano2nnnn  7369  peano5nnnn  7406  peano5nni  8397  1nn  8405  peano2nn  8406  dfuzi  8826  bj-indeq  11470  bj-nntrans  11492
  Copyright terms: Public domain W3C validator