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 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 176   /\ wa 358

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8

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