Theorem pclem6 555

Description: Negation inferred from embedded conjunct.

Assertion

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

Proof of Theorem pclem6

Step	Hyp	Ref	Expression
1		bi1 130	. . . 4 |- ((ph <-> (ps /\ -. ph)) -> (ph -> (ps /\ -. ph)))
2		pm3.27 260	. . . 4 |- ((ps /\ -. ph) -> -. ph)
3	1, 2	syl6 23	. . 3 |- ((ph <-> (ps /\ -. ph)) -> (ph -> -. ph))
4	3	pm2.01d 81	. 2 |- ((ph <-> (ps /\ -. ph)) -> -. ph)
5		bi2 131	. . . . 5 |- ((ph <-> (ps /\ -. ph)) -> ((ps /\ -. ph) -> ph))
6	5	exp3a 292	. . . 4 |- ((ph <-> (ps /\ -. ph)) -> (ps -> (-. ph -> ph)))
7	6	com23 32	. . 3 |- ((ph <-> (ps /\ -. ph)) -> (-. ph -> (ps -> ph)))
8		con3 86	. . 3 |- ((ps -> ph) -> (-. ph -> -. ps))
9	7, 8	syli 52	. 2 |- ((ph <-> (ps /\ -. ph)) -> (-. ph -> -. ps))
10	4, 9	mpd 46	1 |- ((ph <-> (ps /\ -. ph)) -> -. ps)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196

This theorem is referenced by:  nalset 1482

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

This theorem depends on definitions:  df-bi 128  df-an 198

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