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Theorem moi 2920
Description: Equality implied by "at most one". (Contributed by NM, 18-Feb-2006.)
Hypotheses
Ref Expression
moi.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
moi.2  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
Assertion
Ref Expression
moi  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ( ps  /\  ch ) )  ->  A  =  B )
Distinct variable groups:    x, A    x, B    ch, x    ps, x
Allowed substitution hints:    ph( x)    C( x)    D( x)

Proof of Theorem moi
StepHypRef Expression
1 moi.1 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
2 moi.2 . . . . . 6  |-  ( x  =  B  ->  ( ph 
<->  ch ) )
31, 2mob 2919 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) )
43biimprd 158 . . . 4  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ps )  -> 
( ch  ->  A  =  B ) )
543expia 1205 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph )  ->  ( ps 
->  ( ch  ->  A  =  B ) ) )
65impd 254 . 2  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph )  ->  ( ( ps  /\  ch )  ->  A  =  B ) )
763impia 1200 1  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ( ps  /\  ch ) )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353   E*wmo 2027    e. wcel 2148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739
This theorem is referenced by: (None)
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