Theorem pm4.79dc 809

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 640	. . 3 |- ( ( ( ps -> ph ) \/ ( ch -> ph ) ) -> ( ( ps /\ ch ) -> ph ) )
4		simplimdc 757	. . . . . 6 |- (DECID ps -> ( -. ( ps -> ph ) -> ps ) )
5		pm3.3 248	. . . . . 6 |- ( ( ( ps /\ ch ) -> ph ) -> ( ps -> ( ch -> ph ) ) )
6	4, 5	syl9 66	. . . . 5 |- (DECID ps -> ( ( ( ps /\ ch ) -> ph ) -> ( -. ( ps -> ph ) -> ( ch -> ph ) ) ) )
7		dcim 784	. . . . . 6 |- (DECID ps -> (DECID ph -> DECID ( ps -> ph ) ) )
8		pm2.54dc 790	. . . . . 6 |- (DECID ( ps -> ph ) -> ( ( -. ( ps -> ph ) -> ( ch -> ph ) ) -> ( ( ps -> ph ) \/ ( ch -> ph ) ) ) )
9	7, 8	syl6 29	. . . . 5 |- (DECID ps -> (DECID ph -> ( ( -. ( ps -> ph ) -> ( ch -> ph ) ) -> ( ( ps -> ph ) \/ ( ch -> ph ) ) ) ) )
10	6, 9	syl5d 62	. . . 4 |- (DECID ps -> (DECID ph -> ( ( ( ps /\ ch ) -> ph ) -> ( ( ps -> ph ) \/ ( ch -> ph ) ) ) ) )
11	10	imp 115	. . 3 |- ( (DECID ps /\ DECID ph ) -> ( ( ( ps /\ ch ) -> ph ) -> ( ( ps -> ph ) \/ ( ch -> ph ) ) ) )
12	3, 11	impbid2 131	. 2 |- ( (DECID ps /\ DECID ph ) -> ( ( ( ps -> ph ) \/ ( ch -> ph ) ) <-> ( ( ps /\ ch ) -> ph ) ) )
13	12	expcom 109	1 |- (DECID ph -> (DECID ps -> ( ( ( ps -> ph ) \/ ( ch -> ph ) ) <-> ( ( ps /\ ch ) -> ph ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> 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-in1 544  ax-in2 545  ax-io 630

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

This theorem is referenced by: (None)