Theorem pm5.7dc 944

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 943. (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 943	. 2 |- (DECID ch -> ( ( ch \/ ( ph <-> ps ) ) <-> ( ( ch \/ ph ) <-> ( ch \/ ps ) ) ) )
2		orcom 718	. . 3 |- ( ( ch \/ ph ) <-> ( ph \/ ch ) )
3		orcom 718	. . 3 |- ( ( ch \/ ps ) <-> ( ps \/ ch ) )
4	2, 3	bibi12i 228	. 2 |- ( ( ( ch \/ ph ) <-> ( ch \/ ps ) ) <-> ( ( ph \/ ch ) <-> ( ps \/ ch ) ) )
5	1, 4	bitr2di 196	1 |- (DECID ch -> ( ( ( ph \/ ch ) <-> ( ps \/ ch ) ) <-> ( ch \/ ( ph <-> ps ) ) ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 698  DECID wdc 824

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-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-dc 825

This theorem is referenced by: (None)