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

Theorem raleqf 2668
Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
raleq1f.1  |-  F/_ x A
raleq1f.2  |-  F/_ x B
Assertion
Ref Expression
raleqf  |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ph ) )

Proof of Theorem raleqf
StepHypRef Expression
1 raleq1f.1 . . . 4  |-  F/_ x A
2 raleq1f.2 . . . 4  |-  F/_ x B
31, 2nfeq 2327 . . 3  |-  F/ x  A  =  B
4 eleq2 2241 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54imbi1d 231 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  ->  ph )  <->  ( x  e.  B  ->  ph )
) )
63, 5albid 1615 . 2  |-  ( A  =  B  ->  ( A. x ( x  e.  A  ->  ph )  <->  A. x
( x  e.  B  ->  ph ) ) )
7 df-ral 2460 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
8 df-ral 2460 . 2  |-  ( A. x  e.  B  ph  <->  A. x
( x  e.  B  ->  ph ) )
96, 7, 83bitr4g 223 1  |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1351    = wceq 1353    e. wcel 2148   F/_wnfc 2306   A.wral 2455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460
This theorem is referenced by:  raleq  2672  repizf2  4159  ellimc3apf  13762
  Copyright terms: Public domain W3C validator