Theorem pm4.83dc 858

Description: Theorem *4.83 of [WhiteheadRussell] p. 122, for decidable propositions. As with other case elimination theorems, like pm2.61dc 762, it only holds for decidable propositions. (Contributed by Jim Kingdon, 12-May-2018.)

Assertion

Ref	Expression
pm4.83dc	|- (DECID ph -> ( ( ( ph -> ps ) /\ ( -. ph -> ps ) ) <-> ps ) )

Proof of Theorem pm4.83dc

Step	Hyp	Ref	Expression
1		df-dc 743	. . 3 |- (DECID ph <-> ( ph \/ -. ph ) )
2		pm3.44 635	. . . 4 |- ( ( ( ph -> ps ) /\ ( -. ph -> ps ) ) -> ( ( ph \/ -. ph ) -> ps ) )
3	2	com12 27	. . 3 |- ( ( ph \/ -. ph ) -> ( ( ( ph -> ps ) /\ ( -. ph -> ps ) ) -> ps ) )
4	1, 3	sylbi 114	. 2 |- (DECID ph -> ( ( ( ph -> ps ) /\ ( -. ph -> ps ) ) -> ps ) )
5		ax-1 5	. . 3 |- ( ps -> ( ph -> ps ) )
6		ax-1 5	. . 3 |- ( ps -> ( -. ph -> ps ) )
7	5, 6	jca 290	. 2 |- ( ps -> ( ( ph -> ps ) /\ ( -. ph -> ps ) ) )
8	4, 7	impbid1 130	1 |- (DECID ph -> ( ( ( ph -> ps ) /\ ( -. ph -> ps ) ) <-> ps ) )

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-io 630

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

This theorem is referenced by: (None)