Theorem pm4.82 894

Description: Theorem *4.82 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm4.82	|- (((ph -> ps) /\ (ph -> -. ps)) <-> -. ph)

Proof of Theorem pm4.82

Step	Hyp	Ref	Expression
1		pm2.65 164	. . 3 |- ((ph -> ps) -> ((ph -> -. ps) -> -. ph))
2	1	imp 418	. 2 |- (((ph -> ps) /\ (ph -> -. ps)) -> -. ph)
3		pm2.21 100	. . 3 |- (-. ph -> (ph -> ps))
4		pm2.21 100	. . 3 |- (-. ph -> (ph -> -. ps))
5	3, 4	jca 518	. 2 |- (-. ph -> ((ph -> ps) /\ (ph -> -. ps)))
6	2, 5	impbii 180	1 |- (((ph -> ps) /\ (ph -> -. ps)) <-> -. ph)

This proof depends on syntax axioms:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360

This theorem is used by: (None)