Theorem pclem6 1374

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

Step	Hyp	Ref	Expression
1		biimp 118	. . . . 5 |- ( ( ph <-> ( ps /\ -. ph ) ) -> ( ph -> ( ps /\ -. ph ) ) )
2		pm3.4 333	. . . . . 6 |- ( ( ps /\ -. ph ) -> ( ps -> -. ph ) )
3	2	com12 30	. . . . 5 |- ( ps -> ( ( ps /\ -. ph ) -> -. ph ) )
4	1, 3	syl9r 73	. . . 4 |- ( ps -> ( ( ph <-> ( ps /\ -. ph ) ) -> ( ph -> -. ph ) ) )
5		ax-ia3 108	. . . . 5 |- ( ps -> ( -. ph -> ( ps /\ -. ph ) ) )
6		biimpr 130	. . . . 5 |- ( ( ph <-> ( ps /\ -. ph ) ) -> ( ( ps /\ -. ph ) -> ph ) )
7	5, 6	syl9 72	. . . 4 |- ( ps -> ( ( ph <-> ( ps /\ -. ph ) ) -> ( -. ph -> ph ) ) )
8	4, 7	impbidd 127	. . 3 |- ( ps -> ( ( ph <-> ( ps /\ -. ph ) ) -> ( ph <-> -. ph ) ) )
9		pm5.19 706	. . . 4 |- -. ( ph <-> -. ph )
10	9	pm2.21i 646	. . 3 |- ( ( ph <-> -. ph ) -> F. )
11	8, 10	syl6com 35	. 2 |- ( ( ph <-> ( ps /\ -. ph ) ) -> ( ps -> F. ) )
12		dfnot 1371	. 2 |- ( -. ps <-> ( ps -> F. ) )
13	11, 12	sylibr 134	1 |- ( ( ph <-> ( ps /\ -. ph ) ) -> -. ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   F. wfal 1358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359

This theorem is referenced by:  nalset  4132  pwnss  4158  bj-nalset  14498