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Theorem raleqf 3021
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 2595 . . 3  |-  F/ x  A  =  B
4 eleq2 2495 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54imbi1d 318 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  ->  ph )  <->  ( x  e.  B  ->  ph )
) )
63, 5albid 1936 . 2  |-  ( A  =  B  ->  ( A. x ( x  e.  A  ->  ph )  <->  A. x
( x  e.  B  ->  ph ) ) )
7 df-ral 2780 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
8 df-ral 2780 . 2  |-  ( A. x  e.  B  ph  <->  A. x
( x  e.  B  ->  ph ) )
96, 7, 83bitr4g 291 1  |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435    = wceq 1437    e. wcel 1868   F/_wnfc 2570   A.wral 2775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780
This theorem is referenced by:  raleq  3025  raleqbid  3037  dfon2lem3  30426  indexa  31974  ralbi12f  32318  iineq12f  32322  ac6s6f  32330  stoweidlem28  37708  stoweidlem52  37733  fourierdlem31  37820  fourierdlem31OLD  37821  fourierdlem68  37858  fourierdlem103  37893  fourierdlem104  37894
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