Theorem pm5.7dc 861

Description: Disjunction distributes over the biconditional, for a decidable proposition. Based on theorem *5.7 of [WhiteheadRussell] p. 125. This theorem is similar to orbididc 860. (Contributed by Jim Kingdon, 2-Apr-2018.)

Assertion

Ref	Expression
pm5.7dc	|- (DECID ch -> ( ( ( ph \/ ch ) <-> ( ps \/ ch ) ) <-> ( ch \/ ( ph <-> ps ) ) ) )

Proof of Theorem pm5.7dc

Step	Hyp	Ref	Expression
1		orbididc 860	. 2 |- (DECID ch -> ( ( ch \/ ( ph <-> ps ) ) <-> ( ( ch \/ ph ) <-> ( ch \/ ps ) ) ) )
2		orcom 647	. . 3 |- ( ( ch \/ ph ) <-> ( ph \/ ch ) )
3		orcom 647	. . 3 |- ( ( ch \/ ps ) <-> ( ps \/ ch ) )
4	2, 3	bibi12i 218	. 2 |- ( ( ( ch \/ ph ) <-> ( ch \/ ps ) ) <-> ( ( ph \/ ch ) <-> ( ps \/ ch ) ) )
5	1, 4	syl6rbb 186	1 |- (DECID ch -> ( ( ( ph \/ ch ) <-> ( ps \/ ch ) ) <-> ( ch \/ ( ph <-> ps ) ) ) )

Syntax hints:   -> wi 4   <-> wb 98   \/ 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-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by: (None)