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Theorem moi 2840
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 2839 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) )
43biimprd 157 . . . 4  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ps )  -> 
( ch  ->  A  =  B ) )
543expia 1168 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph )  ->  ( ps 
->  ( ch  ->  A  =  B ) ) )
65impd 252 . 2  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph )  ->  ( ( ps  /\  ch )  ->  A  =  B ) )
763impia 1163 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 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   E*wmo 1978
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662
This theorem is referenced by: (None)
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