Theorem jaddc 761

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 48	. 2 |- ( ph -> ( ( ps -> ch ) -> ( ps -> th ) ) )
3		jaddc.1	. . 3 |- ( ph -> (DECID ps -> ( -. ps -> th ) ) )
4		pm2.6dc 759	. . 3 |- (DECID ps -> ( ( -. ps -> th ) -> ( ( ps -> th ) -> th ) ) )
5	3, 4	sylcom 25	. 2 |- ( ph -> (DECID ps -> ( ( ps -> th ) -> th ) ) )
6	2, 5	syl5d 62	1 |- ( ph -> (DECID ps -> ( ( ps -> ch ) -> th ) ) )

Syntax hints:   -. wn 3   -> wi 4  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:  pm2.54dc  790