Theorem pm2.6dc 847

Description: Case elimination for a decidable proposition. Based on theorem *2.6 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 25-Mar-2018.)

Assertion

Ref	Expression
pm2.6dc	|- (DECID ph -> ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) )

Proof of Theorem pm2.6dc

Step	Hyp	Ref	Expression
1		pm2.1dc 822	. . 3 |- (DECID ph -> ( -. ph \/ ph ) )
2		pm3.44 704	. . 3 |- ( ( ( -. ph -> ps ) /\ ( ph -> ps ) ) -> ( ( -. ph \/ ph ) -> ps ) )
3	1, 2	syl5com 29	. 2 |- (DECID ph -> ( ( ( -. ph -> ps ) /\ ( ph -> ps ) ) -> ps ) )
4	3	expd 256	1 |- (DECID ph -> ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697  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-io 698

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

This theorem is referenced by:  jadc  848  jaddc  849  pm2.61dc  850