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Theorem moi2 3108
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 3107 . . . 4  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( x  =  A  <->  ps ) )
323expa 1153 . . 3  |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ph )  ->  (
x  =  A  <->  ps )
)
43biimprd 215 . 2  |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ph )  ->  ( ps  ->  x  =  A ) )
54impr 603 1  |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ( ph  /\  ps ) )  ->  x  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E*wmo 2282
This theorem is referenced by:  fsum  12507  txcn  17651  haustsms2  18159  fprod  25260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2951
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