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Theorem rexeqf 2995
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 2613 . . 3  |-  F/ x  A  =  B
4 eleq2 2528 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54anbi1d 716 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ph )
) )
63, 5exbid 1974 . 2  |-  ( A  =  B  ->  ( E. x ( x  e.  A  /\  ph )  <->  E. x ( x  e.  B  /\  ph )
) )
7 df-rex 2754 . 2  |-  ( E. x  e.  A  ph  <->  E. x ( x  e.  A  /\  ph )
)
8 df-rex 2754 . 2  |-  ( E. x  e.  B  ph  <->  E. x ( x  e.  B  /\  ph )
)
96, 7, 83bitr4g 296 1  |-  ( A  =  B  ->  ( E. x  e.  A  ph  <->  E. x  e.  B  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454   E.wex 1673    e. wcel 1897   F/_wnfc 2589   E.wrex 2749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-cleq 2454  df-clel 2457  df-nfc 2591  df-rex 2754
This theorem is referenced by:  rexeq  2999  rexeqbid  3011  zfrep6  6787  iuneq12daf  28218  indexa  32104
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