Theorem jadc 858

Description: Inference forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 25-Mar-2018.)

Hypotheses

Ref	Expression
jadc.1	|- (DECID ph -> ( -. ph -> ch ) )
jadc.2	|- ( ps -> ch )

Assertion

Ref	Expression
jadc	|- (DECID ph -> ( ( ph -> ps ) -> ch ) )

Proof of Theorem jadc

Step	Hyp	Ref	Expression
1		jadc.2	. . 3 |- ( ps -> ch )
2	1	imim2i 12	. 2 |- ( ( ph -> ps ) -> ( ph -> ch ) )
3		jadc.1	. . 3 |- (DECID ph -> ( -. ph -> ch ) )
4		pm2.6dc 857	. . 3 |- (DECID ph -> ( ( -. ph -> ch ) -> ( ( ph -> ch ) -> ch ) ) )
5	3, 4	mpd 13	. 2 |- (DECID ph -> ( ( ph -> ch ) -> ch ) )
6	2, 5	syl5 32	1 |- (DECID ph -> ( ( ph -> ps ) -> ch ) )

Syntax hints:   -. wn 3   -> wi 4  DECID wdc 829

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 704

This theorem depends on definitions:  df-bi 116  df-dc 830

This theorem is referenced by:  pm5.71dc  956