Theorem pm4.82 969

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		pm2.65 184	. . 3 |- ( ( ph -> ps ) -> ( ( ph -> -. ps ) -> -. ph ) )
2	1	imp 445	. 2 |- ( ( ( ph -> ps ) /\ ( ph -> -. ps ) ) -> -. ph )
3		pm2.21 120	. . 3 |- ( -. ph -> ( ph -> ps ) )
4		pm2.21 120	. . 3 |- ( -. ph -> ( ph -> -. ps ) )
5	3, 4	jca 554	. 2 |- ( -. ph -> ( ( ph -> ps ) /\ ( ph -> -. ps ) ) )
6	2, 5	impbii 199	1 |- ( ( ( ph -> ps ) /\ ( ph -> -. ps ) ) <-> -. ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384

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

This theorem depends on definitions:  df-bi 197  df-an 386

This theorem is referenced by:  alimp-no-surprise  42527