Theorem jaddc 859

Description: Deduction forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 26-Mar-2018.)

Hypotheses

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

Assertion

Ref	Expression
jaddc	|- ( ph -> (DECID ps -> ( ( ps -> ch ) -> th ) ) )

Proof of Theorem jaddc

Step	Hyp	Ref	Expression
1		jaddc.2	. . 3 |- ( ph -> ( ch -> th ) )
2	1	imim2d 54	. 2 |- ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) )
3		jaddc.1	. . 3 |- ( ph -> (DECID ps -> ( -. ps -> th ) ) )
4		pm2.6dc 857	. . 3 |- (DECID ps -> ( ( -. ps -> th ) -> ( ( ps -> th ) -> th ) ) )
5	3, 4	sylcom 28	. 2 |- ( ph -> (DECID ps -> ( ( ps -> th ) -> th ) ) )
6	2, 5	syl5d 68	1 |- ( ph -> (DECID ps -> ( ( ps -> ch ) -> th ) ) )

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:  pm2.54dc  886