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Theorem pclem6 1306
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 116 . . . . 5  |-  ( (
ph 
<->  ( ps  /\  -.  ph ) )  ->  ( ph  ->  ( ps  /\  -.  ph ) ) )
2 pm3.4 326 . . . . . 6  |-  ( ( ps  /\  -.  ph )  ->  ( ps  ->  -. 
ph ) )
32com12 30 . . . . 5  |-  ( ps 
->  ( ( ps  /\  -.  ph )  ->  -.  ph ) )
41, 3syl9r 72 . . . 4  |-  ( ps 
->  ( ( ph  <->  ( ps  /\ 
-.  ph ) )  -> 
( ph  ->  -.  ph ) ) )
5 ax-ia3 106 . . . . 5  |-  ( ps 
->  ( -.  ph  ->  ( ps  /\  -.  ph ) ) )
6 bi2 128 . . . . 5  |-  ( (
ph 
<->  ( ps  /\  -.  ph ) )  ->  (
( ps  /\  -.  ph )  ->  ph ) )
75, 6syl9 71 . . . 4  |-  ( ps 
->  ( ( ph  <->  ( ps  /\ 
-.  ph ) )  -> 
( -.  ph  ->  ph ) ) )
84, 7impbidd 125 . . 3  |-  ( ps 
->  ( ( ph  <->  ( ps  /\ 
-.  ph ) )  -> 
( ph  <->  -.  ph ) ) )
9 pm5.19 655 . . . 4  |-  -.  ( ph 
<->  -.  ph )
109pm2.21i 608 . . 3  |-  ( (
ph 
<->  -.  ph )  -> F.  )
118, 10syl6com 35 . 2  |-  ( (
ph 
<->  ( ps  /\  -.  ph ) )  ->  ( ps  -> F.  ) )
12 dfnot 1303 . 2  |-  ( -. 
ps 
<->  ( ps  -> F.  ) )
1311, 12sylibr 132 1  |-  ( (
ph 
<->  ( ps  /\  -.  ph ) )  ->  -.  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103   F. wfal 1290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291
This theorem is referenced by:  nalset  3928  pwnss  3953  bj-nalset  10953
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