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

Theorem raleqbi1dv 2609
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 2601 . 2  |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ph ) )
2 raleqd.1 . . 3  |-  ( A  =  B  ->  ( ph 
<->  ps ) )
32ralbidv 2412 . 2  |-  ( A  =  B  ->  ( A. x  e.  B  ph  <->  A. x  e.  B  ps ) )
41, 3bitrd 187 1  |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1314   A.wral 2391
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396
This theorem is referenced by:  frforeq2  4235  weeq2  4247  peano5  4480  isoeq4  5671  exmidomni  6980  pitonn  7620  peano1nnnn  7624  peano2nnnn  7625  peano5nnnn  7664  peano5nni  8683  1nn  8691  peano2nn  8692  dfuzi  9115  istopg  12072  isbasisg  12117  basis2  12121  eltg2  12128  ispsmet  12398  ismet  12419  isxmet  12420  metrest  12581  cncfval  12634  bj-indeq  12961  bj-nntrans  12983
  Copyright terms: Public domain W3C validator