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Theorem rmoi2 3374
Description: Consequence of "restricted at most one." (Contributed by Thierry Arnoux, 9-Dec-2019.)
Hypotheses
Ref Expression
rmoi2.1  |-  ( x  =  B  ->  ( ps 
<->  ch ) )
rmoi2.2  |-  ( ph  ->  B  e.  A )
rmoi2.3  |-  ( ph  ->  E* x  e.  A  ps )
rmoi2.4  |-  ( ph  ->  x  e.  A )
rmoi2.5  |-  ( ph  ->  ps )
rmoi2.6  |-  ( ph  ->  ch )
Assertion
Ref Expression
rmoi2  |-  ( ph  ->  x  =  B )
Distinct variable groups:    x, A    x, B    ch, x
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem rmoi2
StepHypRef Expression
1 rmoi2.6 . 2  |-  ( ph  ->  ch )
2 rmoi2.1 . . 3  |-  ( x  =  B  ->  ( ps 
<->  ch ) )
3 rmoi2.2 . . 3  |-  ( ph  ->  B  e.  A )
4 rmoi2.3 . . 3  |-  ( ph  ->  E* x  e.  A  ps )
5 rmoi2.4 . . 3  |-  ( ph  ->  x  e.  A )
6 rmoi2.5 . . 3  |-  ( ph  ->  ps )
72, 3, 4, 5, 6rmob2 3373 . 2  |-  ( ph  ->  ( x  =  B  <->  ch ) )
81, 7mpbird 234 1  |-  ( ph  ->  x  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    = wceq 1407    e. wcel 1844   E*wrmo 2759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-rmo 2764  df-v 3063
This theorem is referenced by:  lmieu  24545
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