Theorem pclem6 813

Description: Negation inferred from embedded conjunct.

Assertion

Ref	Expression
pclem6	|- ((ph <-> (ps /\ -. ph)) -> -. ps)

Proof of Theorem pclem6

Step	Hyp	Ref	Expression
1		bi1 165	. . . 4 |- ((ph <-> (ps /\ -. ph)) -> (ph -> (ps /\ -. ph)))
2		simpr 350	. . . 4 |- ((ps /\ -. ph) -> -. ph)
3	1, 2	syl6 25	. . 3 |- ((ph <-> (ps /\ -. ph)) -> (ph -> -. ph))
4	3	pm2.01d 105	. 2 |- ((ph <-> (ps /\ -. ph)) -> -. ph)
5		bi2 166	. . . . 5 |- ((ph <-> (ps /\ -. ph)) -> ((ps /\ -. ph) -> ph))
6	5	exp3a 405	. . . 4 |- ((ph <-> (ps /\ -. ph)) -> (ps -> (-. ph -> ph)))
7	6	com23 36	. . 3 |- ((ph <-> (ps /\ -. ph)) -> (-. ph -> (ps -> ph)))
8		con3 110	. . 3 |- ((ps -> ph) -> (-. ph -> -. ps))
9	7, 8	syli 65	. 2 |- ((ph <-> (ps /\ -. ph)) -> (-. ph -> -. ps))
10	4, 9	mpd 29	1 |- ((ph <-> (ps /\ -. ph)) -> -. ps)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240

This theorem is referenced by:  nalset 3448

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-an 242

Copyright terms: Public domain