Theorem pclem6 971

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

Step	Hyp	Ref	Expression
1		ibar 525	. . 3 |- ( ps -> ( -. ph <-> ( ps /\ -. ph ) ) )
2		nbbn 373	. . 3 |- ( ( -. ph <-> ( ps /\ -. ph ) ) <-> -. ( ph <-> ( ps /\ -. ph ) ) )
3	1, 2	sylib 208	. 2 |- ( ps -> -. ( ph <-> ( ps /\ -. ph ) ) )
4	3	con2i 134	1 |- ( ( ph <-> ( ps /\ -. ph ) ) -> -. ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386

This theorem is referenced by:  nalset  4795  pwnss  4830  bj-nalset  32794