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Theorem rmoeq1f 3031
Description: Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypotheses
Ref Expression
raleq1f.1  |-  F/_ x A
raleq1f.2  |-  F/_ x B
Assertion
Ref Expression
rmoeq1f  |-  ( A  =  B  ->  ( E* x  e.  A  ph  <->  E* x  e.  B  ph ) )

Proof of Theorem rmoeq1f
StepHypRef Expression
1 raleq1f.1 . . . 4  |-  F/_ x A
2 raleq1f.2 . . . 4  |-  F/_ x B
31, 2nfeq 2602 . . 3  |-  F/ x  A  =  B
4 eleq2 2502 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54anbi1d 709 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ph )
) )
63, 5mobid 2287 . 2  |-  ( A  =  B  ->  ( E* x ( x  e.  A  /\  ph )  <->  E* x ( x  e.  B  /\  ph )
) )
7 df-rmo 2790 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
8 df-rmo 2790 . 2  |-  ( E* x  e.  B  ph  <->  E* x ( x  e.  B  /\  ph )
)
96, 7, 83bitr4g 291 1  |-  ( A  =  B  ->  ( E* x  e.  A  ph  <->  E* x  e.  B  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   E*wmo 2267   F/_wnfc 2577   E*wrmo 2785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-eu 2270  df-mo 2271  df-cleq 2421  df-clel 2424  df-nfc 2579  df-rmo 2790
This theorem is referenced by:  rmoeq1  3035
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