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Theorem raleqf 2622
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 2289 . . 3  |-  F/ x  A  =  B
4 eleq2 2203 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54imbi1d 230 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  ->  ph )  <->  ( x  e.  B  ->  ph )
) )
63, 5albid 1594 . 2  |-  ( A  =  B  ->  ( A. x ( x  e.  A  ->  ph )  <->  A. x
( x  e.  B  ->  ph ) ) )
7 df-ral 2421 . 2  |-  ( A. x  e.  A  ph  <->  A. x
( x  e.  A  ->  ph ) )
8 df-ral 2421 . 2  |-  ( A. x  e.  B  ph  <->  A. x
( x  e.  B  ->  ph ) )
96, 7, 83bitr4g 222 1  |-  ( A  =  B  ->  ( A. x  e.  A  ph  <->  A. x  e.  B  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1329    = wceq 1331    e. wcel 1480   F/_wnfc 2268   A.wral 2416
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421
This theorem is referenced by:  raleq  2626  repizf2  4086  ellimc3apf  12812
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