Theorem pclem6 1264

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		bi1 111	. . . . 5 |- ((ph <-> (ps /\ -. ph)) -> (ph -> (ps /\ -. ph)))
2		pm3.4 316	. . . . . 6 |- ((ps /\ -. ph) -> (ps -> -. ph))
3	2	com12 27	. . . . 5 |- (ps -> ((ps /\ -. ph) -> -. ph))
4	1, 3	syl9r 67	. . . 4 |- (ps -> ((ph <-> (ps /\ -. ph)) -> (ph -> -. ph)))
5		ax-ia3 101	. . . . 5 |- (ps -> (-. ph -> (ps /\ -. ph)))
6		bi2 121	. . . . 5 |- ((ph <-> (ps /\ -. ph)) -> ((ps /\ -. ph) -> ph))
7	5, 6	syl9 66	. . . 4 |- (ps -> ((ph <-> (ps /\ -. ph)) -> (-. ph -> ph)))
8	4, 7	impbidd 118	. . 3 |- (ps -> ((ph <-> (ps /\ -. ph)) -> (ph <-> -. ph)))
9		pm5.19 621	. . . 4 |- -. (ph <-> -. ph)
10	9	pm2.21i 574	. . 3 |- ((ph <-> -. ph) -> F. )
11	8, 10	syl6com 31	. 2 |- ((ph <-> (ps /\ -. ph)) -> (ps -> F. ))
12		dfnot 1261	. 2 |- (-. ps <-> (ps -> F. ))
13	11, 12	sylibr 137	1 |- ((ph <-> (ps /\ -. ph)) -> -. ps)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   F. wfal 1247

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248

This theorem is referenced by:  nalset  3877  pwnss  3902  bj-nalset  8945