Theorem pm5.12dc 906

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 831	. 2 |- (DECID ps <-> ( ps \/ -. ps ) )
2		ax-1 6	. . 3 |- ( ps -> ( ph -> ps ) )
3		ax-1 6	. . 3 |- ( -. ps -> ( ph -> -. ps ) )
4	2, 3	orim12i 755	. 2 |- ( ( ps \/ -. ps ) -> ( ( ph -> ps ) \/ ( ph -> -. ps ) ) )
5	1, 4	sylbi 120	1 |- (DECID ps -> ( ( ph -> ps ) \/ ( ph -> -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 704  DECID wdc 830

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 705

This theorem depends on definitions:  df-bi 116  df-dc 831

This theorem is referenced by: (None)