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Theorem rexeqf 2837
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 2523 . . 3  |-  F/ x  A  =  B
4 eleq2 2441 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54anbi1d 686 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ph )
) )
63, 5exbid 1781 . 2  |-  ( A  =  B  ->  ( E. x ( x  e.  A  /\  ph )  <->  E. x ( x  e.  B  /\  ph )
) )
7 df-rex 2648 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
8 df-rex 2648 . 2  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
96, 7, 83bitr4g 280 1  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   F/_wnfc 2503   E.wrex 2643
This theorem is referenced by:  rexeq  2841  zfrep6  5900  rexeqbid  23801  iuneq12daf  23844  indexa  26119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rex 2648
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