Theorem pm2.6dc 798

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 784	. . 3 |- (DECID ph -> ( -. ph \/ ph ) )
2		pm3.44 671	. . 3 |- ( ( ( -. ph -> ps ) /\ ( ph -> ps ) ) -> ( ( -. ph \/ ph ) -> ps ) )
3	1, 2	syl5com 29	. 2 |- (DECID ph -> ( ( ( -. ph -> ps ) /\ ( ph -> ps ) ) -> ps ) )
4	3	expd 255	1 |- (DECID ph -> ( ( -. ph -> ps ) -> ( ( ph -> ps ) -> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 665  DECID wdc 781

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666

This theorem depends on definitions:  df-bi 116  df-dc 782

This theorem is referenced by:  jadc  799  jaddc  800  pm2.61dc  801