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Theorem rmoeq1f 2561
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 2236 . . 3  |-  F/ x  A  =  B
4 eleq2 2151 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
54anbi1d 453 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  ph )  <->  ( x  e.  B  /\  ph )
) )
63, 5mobid 1983 . 2  |-  ( A  =  B  ->  ( E* x ( x  e.  A  /\  ph )  <->  E* x ( x  e.  B  /\  ph )
) )
7 df-rmo 2367 . 2  |-  ( E* x  e.  A  ph  <->  E* x ( x  e.  A  /\  ph )
)
8 df-rmo 2367 . 2  |-  ( E* x  e.  B  ph  <->  E* x ( x  e.  B  /\  ph )
)
96, 7, 83bitr4g 221 1  |-  ( A  =  B  ->  ( E* x  e.  A  ph  <->  E* x  e.  B  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   E*wmo 1949   F/_wnfc 2215   E*wrmo 2362
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rmo 2367
This theorem is referenced by:  rmoeq1  2565
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