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Theorem conimpf 27989
Description: Assuming a, not b, and a implies b, there exists a proof that a is false.) (Contributed by Jarvin Udandy, 28-Aug-2016.)
Hypotheses
Ref Expression
conimpf.1  |-  ph
conimpf.2  |-  -.  ps
conimpf.3  |-  ( ph  ->  ps )
Assertion
Ref Expression
conimpf  |-  ( ph  <->  F.  )

Proof of Theorem conimpf
StepHypRef Expression
1 conimpf.1 . . . 4  |-  ph
2 conimpf.3 . . . . 5  |-  ( ph  ->  ps )
3 atbiffatnnb 27984 . . . . 5  |-  ( (
ph  ->  ps )  -> 
( ph  ->  -.  -.  ps ) )
42, 3ax-mp 8 . . . 4  |-  ( ph  ->  -.  -.  ps )
51, 4ax-mp 8 . . 3  |-  -.  -.  ps
6 notnot2 104 . . 3  |-  ( -. 
-.  ps  ->  ps )
75, 6ax-mp 8 . 2  |-  ps
8 conimpf.2 . . 3  |-  -.  ps
9 pm2.21 100 . . 3  |-  ( -. 
ps  ->  ( ps  ->  (
ph 
<->  F.  ) ) )
108, 9ax-mp 8 . 2  |-  ( ps 
->  ( ph  <->  F.  )
)
117, 10ax-mp 8 1  |-  ( ph  <->  F.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    F. wfal 1308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177
  Copyright terms: Public domain W3C validator