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Theorem moi 3081
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 3080 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) )
43biimprd 215 . . . 4  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ps )  -> 
( ch  ->  A  =  B ) )
543expia 1155 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph )  ->  ( ps 
->  ( ch  ->  A  =  B ) ) )
65imp3a 421 . 2  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph )  ->  ( ( ps  /\  ch )  ->  A  =  B ) )
763impia 1150 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E*wmo 2259
This theorem is referenced by:  enqeq  8771  hausflim  17970  f1otrspeq  27262
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-v 2922
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