Theorem pclem6 741

Description: Negation inferred from embedded conjunct.

Assertion

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

Proof of Theorem pclem6

Step	Hyp	Ref	Expression
1		bi1 148	. . . 4 |- ((ph <-> (ps /\ -. ph)) -> (ph -> (ps /\ -. ph)))
2		pm3.27 323	. . . 4 |- ((ps /\ -. ph) -> -. ph)
3	1, 2	syl6 22	. . 3 |- ((ph <-> (ps /\ -. ph)) -> (ph -> -. ph))
4	3	pm2.01d 89	. 2 |- ((ph <-> (ps /\ -. ph)) -> -. ph)
5		bi2 149	. . . . 5 |- ((ph <-> (ps /\ -. ph)) -> ((ps /\ -. ph) -> ph))
6	5	exp3a 375	. . . 4 |- ((ph <-> (ps /\ -. ph)) -> (ps -> (-. ph -> ph)))
7	6	com23 32	. . 3 |- ((ph <-> (ps /\ -. ph)) -> (-. ph -> (ps -> ph)))
8		con3 94	. . 3 |- ((ps -> ph) -> (-. ph -> -. ps))
9	7, 8	syli 54	. 2 |- ((ph <-> (ps /\ -. ph)) -> (-. ph -> -. ps))
10	4, 9	mpd 26	1 |- ((ph <-> (ps /\ -. ph)) -> -. ps)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223

This theorem is referenced by:  nalset 2712

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 147  df-an 225

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