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Theorem pclem6 896
Description: Negation inferred from embedded conjunct. (Contributed by NM, 20-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Nov-2012.)
Assertion
Ref Expression
pclem6  |-  ( (
ph 
<->  ( ps  /\  -.  ph ) )  ->  -.  ps )

Proof of Theorem pclem6
StepHypRef Expression
1 ibar 490 . . 3  |-  ( ps 
->  ( -.  ph  <->  ( ps  /\ 
-.  ph ) ) )
2 nbbn 347 . . 3  |-  ( ( -.  ph  <->  ( ps  /\  -.  ph ) )  <->  -.  ( ph 
<->  ( ps  /\  -.  ph ) ) )
31, 2sylib 188 . 2  |-  ( ps 
->  -.  ( ph  <->  ( ps  /\ 
-.  ph ) ) )
43con2i 112 1  |-  ( (
ph 
<->  ( ps  /\  -.  ph ) )  ->  -.  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358
This theorem is referenced by:  nalset  4151  pwnss  4176
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  df-an 360
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