Theorem orbididc 937

Description: Disjunction distributes over the biconditional, for a decidable proposition. Based on an axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384. (Contributed by Jim Kingdon, 2-Apr-2018.)

Assertion

Ref	Expression
orbididc	|- (DECID ph -> ( ( ph \/ ( ps <-> ch ) ) <-> ( ( ph \/ ps ) <-> ( ph \/ ch ) ) ) )

Proof of Theorem orbididc

Step	Hyp	Ref	Expression
1		orimdidc 891	. . 3 |- (DECID ph -> ( ( ph \/ ( ps -> ch ) ) <-> ( ( ph \/ ps ) -> ( ph \/ ch ) ) ) )
2		orimdidc 891	. . 3 |- (DECID ph -> ( ( ph \/ ( ch -> ps ) ) <-> ( ( ph \/ ch ) -> ( ph \/ ps ) ) ) )
3	1, 2	anbi12d 464	. 2 |- (DECID ph -> ( ( ( ph \/ ( ps -> ch ) ) /\ ( ph \/ ( ch -> ps ) ) ) <-> ( ( ( ph \/ ps ) -> ( ph \/ ch ) ) /\ ( ( ph \/ ch ) -> ( ph \/ ps ) ) ) ) )
4		dfbi2 385	. . . 4 |- ( ( ps <-> ch ) <-> ( ( ps -> ch ) /\ ( ch -> ps ) ) )
5	4	orbi2i 751	. . 3 |- ( ( ph \/ ( ps <-> ch ) ) <-> ( ph \/ ( ( ps -> ch ) /\ ( ch -> ps ) ) ) )
6		ordi 805	. . 3 |- ( ( ph \/ ( ( ps -> ch ) /\ ( ch -> ps ) ) ) <-> ( ( ph \/ ( ps -> ch ) ) /\ ( ph \/ ( ch -> ps ) ) ) )
7	5, 6	bitri 183	. 2 |- ( ( ph \/ ( ps <-> ch ) ) <-> ( ( ph \/ ( ps -> ch ) ) /\ ( ph \/ ( ch -> ps ) ) ) )
8		dfbi2 385	. 2 |- ( ( ( ph \/ ps ) <-> ( ph \/ ch ) ) <-> ( ( ( ph \/ ps ) -> ( ph \/ ch ) ) /\ ( ( ph \/ ch ) -> ( ph \/ ps ) ) ) )
9	3, 7, 8	3bitr4g 222	1 |- (DECID ph -> ( ( ph \/ ( ps <-> ch ) ) <-> ( ( ph \/ ps ) <-> ( ph \/ ch ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697  DECID wdc 819

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 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-dc 820

This theorem is referenced by:  pm5.7dc  938