Theorem pm4.79dc 847

Description: Equivalence between a disjunction of two implications, and a conjunction and an implication. Based on theorem *4.79 of [WhiteheadRussell] p. 121 but with additional decidability antecedents. (Contributed by Jim Kingdon, 28-Mar-2018.)

Assertion

Ref	Expression
pm4.79dc	|- (DECID ph -> (DECID ps -> ( ( ( ps -> ph ) \/ ( ch -> ph ) ) <-> ( ( ps /\ ch ) -> ph ) ) ) )

Proof of Theorem pm4.79dc

Step	Hyp	Ref	Expression
1		id 19	. . . 4 |- ( ( ps -> ph ) -> ( ps -> ph ) )
2		id 19	. . . 4 |- ( ( ch -> ph ) -> ( ch -> ph ) )
3	1, 2	jaoa 675	. . 3 |- ( ( ( ps -> ph ) \/ ( ch -> ph ) ) -> ( ( ps /\ ch ) -> ph ) )
4		simplimdc 795	. . . . . 6 |- (DECID ps -> ( -. ( ps -> ph ) -> ps ) )
5		pm3.3 257	. . . . . 6 |- ( ( ( ps /\ ch ) -> ph ) -> ( ps -> ( ch -> ph ) ) )
6	4, 5	syl9 71	. . . . 5 |- (DECID ps -> ( ( ( ps /\ ch ) -> ph ) -> ( -. ( ps -> ph ) -> ( ch -> ph ) ) ) )
7		dcim 822	. . . . . 6 |- (DECID ps -> (DECID ph -> DECID ( ps -> ph ) ) )
8		pm2.54dc 828	. . . . . 6 |- (DECID ( ps -> ph ) -> ( ( -. ( ps -> ph ) -> ( ch -> ph ) ) -> ( ( ps -> ph ) \/ ( ch -> ph ) ) ) )
9	7, 8	syl6 33	. . . . 5 |- (DECID ps -> (DECID ph -> ( ( -. ( ps -> ph ) -> ( ch -> ph ) ) -> ( ( ps -> ph ) \/ ( ch -> ph ) ) ) ) )
10	6, 9	syl5d 67	. . . 4 |- (DECID ps -> (DECID ph -> ( ( ( ps /\ ch ) -> ph ) -> ( ( ps -> ph ) \/ ( ch -> ph ) ) ) ) )
11	10	imp 122	. . 3 |- ( (DECID ps /\ DECID ph ) -> ( ( ( ps /\ ch ) -> ph ) -> ( ( ps -> ph ) \/ ( ch -> ph ) ) ) )
12	3, 11	impbid2 141	. 2 |- ( (DECID ps /\ DECID ph ) -> ( ( ( ps -> ph ) \/ ( ch -> ph ) ) <-> ( ( ps /\ ch ) -> ph ) ) )
13	12	expcom 114	1 |- (DECID ph -> (DECID ps -> ( ( ( ps -> ph ) \/ ( ch -> ph ) ) <-> ( ( ps /\ ch ) -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 664  DECID wdc 780

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-in1 579  ax-in2 580  ax-io 665

This theorem depends on definitions:  df-bi 115  df-dc 781

This theorem is referenced by: (None)