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Theorem rexeqf 2502
Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
raleq1f.1  |-  F/_ x A
raleq1f.2  |-  F/_ x B
Assertion
Ref Expression
rexeqf  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )

Proof of Theorem rexeqf
StepHypRef Expression
1 raleq1f.1 . . . 4  |-  F/_ x A
2 raleq1f.2 . . . 4  |-  F/_ x B
31, 2nfeq 2185 . . 3  |-  F/ x  A  =  B
4 eleq2 2101 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54anbi1d 438 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ph )
) )
63, 5exbid 1507 . 2  |-  ( A  =  B  ->  ( E. x ( x  e.  A  /\  ph )  <->  E. x ( x  e.  B  /\  ph )
) )
7 df-rex 2312 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
8 df-rex 2312 . 2  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
96, 7, 83bitr4g 212 1  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243   E.wex 1381    e. wcel 1393   F/_wnfc 2165   E.wrex 2307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312
This theorem is referenced by:  rexeq  2506
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