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Theorem moi2 2946
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 2945 . . . 4  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( x  =  A  <->  ps ) )
323expa 1151 . . 3  |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ph )  ->  (
x  =  A  <->  ps )
)
43biimprd 214 . 2  |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ph )  ->  ( ps  ->  x  =  A ) )
54impr 602 1  |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ( ph  /\  ps ) )  ->  x  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E*wmo 2144
This theorem is referenced by:  fsum  12193  txcn  17320  haustsms2  17819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790
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