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Theorem reueq1f 2902
Description: Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
raleq1f.1  |-  F/_ x A
raleq1f.2  |-  F/_ x B
Assertion
Ref Expression
reueq1f  |-  ( A  =  B  ->  ( E! x  e.  A  ph  <->  E! x  e.  B  ph ) )

Proof of Theorem reueq1f
StepHypRef Expression
1 raleq1f.1 . . . 4  |-  F/_ x A
2 raleq1f.2 . . . 4  |-  F/_ x B
31, 2nfeq 2579 . . 3  |-  F/ x  A  =  B
4 eleq2 2497 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54anbi1d 686 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ph )
) )
63, 5eubid 2288 . 2  |-  ( A  =  B  ->  ( E! x ( x  e.  A  /\  ph )  <->  E! x ( x  e.  B  /\  ph )
) )
7 df-reu 2712 . 2  |-  ( E! x  e.  A  ph  <->  E! x ( x  e.  A  /\  ph )
)
8 df-reu 2712 . 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    = wceq 1652    e. wcel 1725   E!weu 2281   F/_wnfc 2559   E!wreu 2707
This theorem is referenced by:  reueq1  2906
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-cleq 2429  df-clel 2432  df-nfc 2561  df-reu 2712
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