Theorem pm4.14dc 875

Description: Theorem *4.14 of [WhiteheadRussell] p. 117, given a decidability condition. (Contributed by Jim Kingdon, 24-Apr-2018.)

Assertion

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

Proof of Theorem pm4.14dc

Step	Hyp	Ref	Expression
1		con34bdc 856	. . 3 |- (DECID ch -> ( ( ps -> ch ) <-> ( -. ch -> -. ps ) ) )
2	1	imbi2d 229	. 2 |- (DECID ch -> ( ( ph -> ( ps -> ch ) ) <-> ( ph -> ( -. ch -> -. ps ) ) ) )
3		impexp 261	. 2 |- ( ( ( ph /\ ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) )
4		impexp 261	. 2 |- ( ( ( ph /\ -. ch ) -> -. ps ) <-> ( ph -> ( -. ch -> -. ps ) ) )
5	2, 3, 4	3bitr4g 222	1 |- (DECID ch -> ( ( ( ph /\ ps ) -> ch ) <-> ( ( ph /\ -. ch ) -> -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820

This theorem is referenced by: (None)