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Theorem raleqf 3054
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 2640 . . 3  |-  F/ x  A  =  B
4 eleq2 2540 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54imbi1d 317 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  ->  ph )  <->  ( x  e.  B  ->  ph )
) )
63, 5albid 1833 . 2  |-  ( A  =  B  ->  ( A. x ( x  e.  A  ->  ph )  <->  A. x
( x  e.  B  ->  ph ) ) )
7 df-ral 2819 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
8 df-ral 2819 . 2  |-  ( A. x  e.  B  ph  <->  A. x
( x  e.  B  ->  ph ) )
96, 7, 83bitr4g 288 1  |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377    = wceq 1379    e. wcel 1767   F/_wnfc 2615   A.wral 2814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819
This theorem is referenced by:  raleq  3058  raleqbid  3070  dfon2lem3  28794  indexa  29827  ralbi12f  30173  iineq12f  30177  ac6s6f  30185  stoweidlem28  31328  stoweidlem52  31352  fourierdlem31  31438  fourierdlem68  31475  fourierdlem103  31510  fourierdlem104  31511
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