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Theorem rmoeq1f 2651
Description: Equality theorem for restricted at-most-one 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 2307 . . 3  |-  F/ x  A  =  B
4 eleq2 2221 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54anbi1d 461 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ph )
) )
63, 5mobid 2041 . 2  |-  ( A  =  B  ->  ( E* x ( x  e.  A  /\  ph )  <->  E* x ( x  e.  B  /\  ph )
) )
7 df-rmo 2443 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
8 df-rmo 2443 . 2  |-  ( E* x  e.  B  ph  <->  E* x ( x  e.  B  /\  ph )
)
96, 7, 83bitr4g 222 1  |-  ( A  =  B  ->  ( E* x  e.  A  ph  <->  E* x  e.  B  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335   E*wmo 2007    e. wcel 2128   F/_wnfc 2286   E*wrmo 2438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rmo 2443
This theorem is referenced by:  rmoeq1  2655
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