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Theorem moi 3119
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 3118 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) )
43biimprd 216 . . . 4  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ps )  -> 
( ch  ->  A  =  B ) )
543expia 1156 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph )  ->  ( ps 
->  ( ch  ->  A  =  B ) ) )
65imp3a 422 . 2  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph )  ->  ( ( ps  /\  ch )  ->  A  =  B ) )
763impia 1151 1  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ( ps  /\  ch ) )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E*wmo 2284
This theorem is referenced by:  enqeq  8816  hausflim  18018  f1otrspeq  27381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960
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