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Theorem pclem6 1335
Description: Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof rewritten by Jim Kingdon, 4-May-2018.)
Assertion
Ref Expression
pclem6  |-  ( (
ph 
<->  ( ps  /\  -.  ph ) )  ->  -.  ps )

Proof of Theorem pclem6
StepHypRef Expression
1 bi1 117 . . . . 5  |-  ( (
ph 
<->  ( ps  /\  -.  ph ) )  ->  ( ph  ->  ( ps  /\  -.  ph ) ) )
2 pm3.4 329 . . . . . 6  |-  ( ( ps  /\  -.  ph )  ->  ( ps  ->  -. 
ph ) )
32com12 30 . . . . 5  |-  ( ps 
->  ( ( ps  /\  -.  ph )  ->  -.  ph ) )
41, 3syl9r 73 . . . 4  |-  ( ps 
->  ( ( ph  <->  ( ps  /\ 
-.  ph ) )  -> 
( ph  ->  -.  ph ) ) )
5 ax-ia3 107 . . . . 5  |-  ( ps 
->  ( -.  ph  ->  ( ps  /\  -.  ph ) ) )
6 bi2 129 . . . . 5  |-  ( (
ph 
<->  ( ps  /\  -.  ph ) )  ->  (
( ps  /\  -.  ph )  ->  ph ) )
75, 6syl9 72 . . . 4  |-  ( ps 
->  ( ( ph  <->  ( ps  /\ 
-.  ph ) )  -> 
( -.  ph  ->  ph ) ) )
84, 7impbidd 126 . . 3  |-  ( ps 
->  ( ( ph  <->  ( ps  /\ 
-.  ph ) )  -> 
( ph  <->  -.  ph ) ) )
9 pm5.19 678 . . . 4  |-  -.  ( ph 
<->  -.  ph )
109pm2.21i 618 . . 3  |-  ( (
ph 
<->  -.  ph )  -> F.  )
118, 10syl6com 35 . 2  |-  ( (
ph 
<->  ( ps  /\  -.  ph ) )  ->  ( ps  -> F.  ) )
12 dfnot 1332 . 2  |-  ( -. 
ps 
<->  ( ps  -> F.  ) )
1311, 12sylibr 133 1  |-  ( (
ph 
<->  ( ps  /\  -.  ph ) )  ->  -.  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104   F. wfal 1319
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-in1 586  ax-in2 587
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320
This theorem is referenced by:  nalset  4026  pwnss  4051  bj-nalset  12904
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