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Theorem moi 3286
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 3285 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ps )  -> 
( A  =  B  <->  ch ) )
43biimprd 223 . . . 4  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ps )  -> 
( ch  ->  A  =  B ) )
543expia 1198 . . 3  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph )  ->  ( ps 
->  ( ch  ->  A  =  B ) ) )
65impd 431 . 2  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph )  ->  ( ( ps  /\  ch )  ->  A  =  B ) )
763impia 1193 1  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  E* x ph  /\  ( ps  /\  ch ) )  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E*wmo 2276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115
This theorem is referenced by:  enqeq  9312  f1otrspeq  16278  hausflim  20245  tglineineq  23764  tglineinteq  23766
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