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Theorem moi2 2865
Description: Consequence of "at most one." (Contributed by NM, 29-Jun-2008.)
Hypothesis
Ref Expression
moi2.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
moi2  |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ( ph  /\  ps ) )  ->  x  =  A )
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem moi2
StepHypRef Expression
1 moi2.1 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21mob2 2864 . . . 4  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( x  =  A  <->  ps ) )
323expa 1181 . . 3  |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ph )  ->  (
x  =  A  <->  ps )
)
43biimprd 157 . 2  |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ph )  ->  ( ps  ->  x  =  A ) )
54impr 376 1  |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ( ph  /\  ps ) )  ->  x  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   E*wmo 2000
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688
This theorem is referenced by:  fsum3  11163  txcn  12454
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