Theorem pm4.14 602

Description: Theorem *4.14 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)

Assertion

Ref	Expression
pm4.14	|- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ph /\ -. ch ) -> -. ps ) )

Proof of Theorem pm4.14

Step	Hyp	Ref	Expression
1		con34b 306	. . 3 |- ( ( ps -> ch ) <-> ( -. ch -> -. ps ) )
2	1	imbi2i 326	. 2 |- ( ( ph -> ( ps -> ch ) ) <-> ( ph -> ( -. ch -> -. ps ) ) )
3		impexp 462	. 2 |- ( ( ( ph /\ ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) )
4		impexp 462	. 2 |- ( ( ( ph /\ -. ch ) -> -. ps ) <-> ( ph -> ( -. ch -> -. ps ) ) )
5	2, 3, 4	3bitr4i 292	1 |- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ph /\ -. ch ) -> -. ps ) )

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:  pm3.37  603  ndvdssub  15133