Theorem pm5.55dc 852

Description: A disjunction is equivalent to one of its disjuncts, given a decidable disjunct. Based on theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)

Assertion

Ref	Expression
pm5.55dc	|- (DECID ph -> ( ( ( ph \/ ps ) <-> ph ) \/ ( ( ph \/ ps ) <-> ps ) ) )

Proof of Theorem pm5.55dc

Step	Hyp	Ref	Expression
1		df-dc 776	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		biort 771	. . . 4 |- ( ph -> ( ph <-> ( ph \/ ps ) ) )
3	2	bicomd 139	. . 3 |- ( ph -> ( ( ph \/ ps ) <-> ph ) )
4		biorf 695	. . . 4 |- ( -. ph -> ( ps <-> ( ph \/ ps ) ) )
5	4	bicomd 139	. . 3 |- ( -. ph -> ( ( ph \/ ps ) <-> ps ) )
6	3, 5	orim12i 708	. 2 |- ( ( ph \/ -. ph ) -> ( ( ( ph \/ ps ) <-> ph ) \/ ( ( ph \/ ps ) <-> ps ) ) )
7	1, 6	sylbi 119	1 |- (DECID ph -> ( ( ( ph \/ ps ) <-> ph ) \/ ( ( ph \/ ps ) <-> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 661  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by: (None)