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Theorem rmoeq1f 2905
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 2581 . . 3  |-  F/ x  A  =  B
4 eleq2 2499 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54anbi1d 687 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ph )
) )
63, 5mobid 2317 . 2  |-  ( A  =  B  ->  ( E* x ( x  e.  A  /\  ph )  <->  E* x ( x  e.  B  /\  ph )
) )
7 df-rmo 2715 . 2  |-  ( E* x  e.  A ph  <->  E* x ( x  e.  A  /\  ph )
)
8 df-rmo 2715 . 2  |-  ( E* x  e.  B ph  <->  E* x ( x  e.  B  /\  ph )
)
96, 7, 83bitr4g 281 1  |-  ( A  =  B  ->  ( E* x  e.  A ph 
<->  E* x  e.  B ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E*wmo 2284   F/_wnfc 2561   E*wrmo 2710
This theorem is referenced by:  rmoeq1  2909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rmo 2715
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