Theorem pm5.12dc 816

Description: Excluded middle with antecedents for a decidable consequent. Based on theorem *5.12 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 30-Mar-2018.)

Assertion

Ref	Expression
pm5.12dc	|- (DECID ps -> ( ( ph -> ps ) \/ ( ph -> -. ps ) ) )

Proof of Theorem pm5.12dc

Step	Hyp	Ref	Expression
1		df-dc 743	. 2 |- (DECID ps <-> ( ps \/ -. ps ) )
2		ax-1 5	. . 3 |- ( ps -> ( ph -> ps ) )
3		ax-1 5	. . 3 |- ( -. ps -> ( ph -> -. ps ) )
4	2, 3	orim12i 676	. 2 |- ( ( ps \/ -. ps ) -> ( ( ph -> ps ) \/ ( ph -> -. ps ) ) )
5	1, 4	sylbi 114	1 |- (DECID ps -> ( ( ph -> ps ) \/ ( ph -> -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 629  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: (None)