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Theorem moi2 3231
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 3230 . . . 4  |-  ( ( A  e.  B  /\  E* x ph  /\  ph )  ->  ( x  =  A  <->  ps ) )
323expa 1215 . . 3  |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ph )  ->  (
x  =  A  <->  ps )
)
43biimprd 231 . 2  |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ph )  ->  ( ps  ->  x  =  A ) )
54impr 629 1  |-  ( ( ( A  e.  B  /\  E* x ph )  /\  ( ph  /\  ps ) )  ->  x  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   E*wmo 2311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059
This theorem is referenced by:  fsum  13835  fprod  14044  txcn  20690  haustsms2  21200
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