Theorem pm2.61ddc 856

Description: Deduction eliminating a decidable antecedent. (Contributed by Jim Kingdon, 4-May-2018.)

Hypotheses

Ref	Expression
pm2.61ddc.1	|- ( ph -> ( ps -> ch ) )
pm2.61ddc.2	|- ( ph -> ( -. ps -> ch ) )

Assertion

Ref	Expression
pm2.61ddc	|- (DECID ps -> ( ph -> ch ) )

Proof of Theorem pm2.61ddc

Step	Hyp	Ref	Expression
1		df-dc 830	. 2 |- (DECID ps <-> ( ps \/ -. ps ) )
2		pm2.61ddc.1	. . . 4 |- ( ph -> ( ps -> ch ) )
3	2	com12 30	. . 3 |- ( ps -> ( ph -> ch ) )
4		pm2.61ddc.2	. . . 4 |- ( ph -> ( -. ps -> ch ) )
5	4	com12 30	. . 3 |- ( -. ps -> ( ph -> ch ) )
6	3, 5	jaoi 711	. 2 |- ( ( ps \/ -. ps ) -> ( ph -> ch ) )
7	1, 6	sylbi 120	1 |- (DECID ps -> ( ph -> ch ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 703  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:  bijadc  877