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Theorem raleqf 2983
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 2603 . . 3  |-  F/ x  A  =  B
4 eleq2 2518 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54imbi1d 319 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  ->  ph )  <->  ( x  e.  B  ->  ph )
) )
63, 5albid 1963 . 2  |-  ( A  =  B  ->  ( A. x ( x  e.  A  ->  ph )  <->  A. x
( x  e.  B  ->  ph ) ) )
7 df-ral 2742 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
8 df-ral 2742 . 2  |-  ( A. x  e.  B  ph  <->  A. x
( x  e.  B  ->  ph ) )
96, 7, 83bitr4g 292 1  |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1442    = wceq 1444    e. wcel 1887   F/_wnfc 2579   A.wral 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742
This theorem is referenced by:  raleq  2987  raleqbid  2999  dfon2lem3  30431  indexa  32060  ralbi12f  32404  iineq12f  32408  ac6s6f  32416  stoweidlem28  37888  stoweidlem52  37913  fourierdlem31  38000  fourierdlem31OLD  38001  fourierdlem68  38038  fourierdlem103  38073  fourierdlem104  38074
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