Theorem pm5.71dc 868

Description: Decidable proposition version of theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.) (Modified for decidability by Jim Kingdon, 19-Apr-2018.)

Assertion

Ref	Expression
pm5.71dc	|- (DECID ps -> ( ( ps -> -. ch ) -> ( ( ( ph \/ ps ) /\ ch ) <-> ( ph /\ ch ) ) ) )

Proof of Theorem pm5.71dc

Step	Hyp	Ref	Expression
1		orel2 645	. . . . 5 |- ( -. ps -> ( ( ph \/ ps ) -> ph ) )
2		orc 633	. . . . 5 |- ( ph -> ( ph \/ ps ) )
3	1, 2	impbid1 130	. . . 4 |- ( -. ps -> ( ( ph \/ ps ) <-> ph ) )
4	3	anbi1d 438	. . 3 |- ( -. ps -> ( ( ( ph \/ ps ) /\ ch ) <-> ( ph /\ ch ) ) )
5	4	a1i 9	. 2 |- (DECID ps -> ( -. ps -> ( ( ( ph \/ ps ) /\ ch ) <-> ( ph /\ ch ) ) ) )
6		pm2.21 547	. . 3 |- ( -. ch -> ( ch -> ( ( ph \/ ps ) <-> ph ) ) )
7	6	pm5.32rd 424	. 2 |- ( -. ch -> ( ( ( ph \/ ps ) /\ ch ) <-> ( ph /\ ch ) ) )
8	5, 7	jadc 760	1 |- (DECID ps -> ( ( ps -> -. ch ) -> ( ( ( ph \/ ps ) /\ ch ) <-> ( ph /\ ch ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by: (None)