Theorem bilukdc 1407

Description: Lukasiewicz's shortest axiom for equivalential calculus (but modified to require decidable propositions). Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96. (Contributed by Jim Kingdon, 5-May-2018.)

Assertion

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

Proof of Theorem bilukdc

Step	Hyp	Ref	Expression
1		bicom 140	. . . . . 6 |- ( ( ph <-> ps ) <-> ( ps <-> ph ) )
2	1	bibi1i 228	. . . . 5 |- ( ( ( ph <-> ps ) <-> ch ) <-> ( ( ps <-> ph ) <-> ch ) )
3		biassdc 1406	. . . . . 6 |- (DECID ps -> (DECID ph -> (DECID ch -> ( ( ( ps <-> ph ) <-> ch ) <-> ( ps <-> ( ph <-> ch ) ) ) ) ) )
4	3	imp31 256	. . . . 5 |- ( ( (DECID ps /\ DECID ph ) /\ DECID ch ) -> ( ( ( ps <-> ph ) <-> ch ) <-> ( ps <-> ( ph <-> ch ) ) ) )
5	2, 4	bitrid 192	. . . 4 |- ( ( (DECID ps /\ DECID ph ) /\ DECID ch ) -> ( ( ( ph <-> ps ) <-> ch ) <-> ( ps <-> ( ph <-> ch ) ) ) )
6	5	ancom1s 569	. . 3 |- ( ( (DECID ph /\ DECID ps ) /\ DECID ch ) -> ( ( ( ph <-> ps ) <-> ch ) <-> ( ps <-> ( ph <-> ch ) ) ) )
7		dcbi 938	. . . . . 6 |- (DECID ph -> (DECID ps -> DECID ( ph <-> ps ) ) )
8	7	imp 124	. . . . 5 |- ( (DECID ph /\ DECID ps ) -> DECID ( ph <-> ps ) )
9	8	adantr 276	. . . 4 |- ( ( (DECID ph /\ DECID ps ) /\ DECID ch ) -> DECID ( ph <-> ps ) )
10		simpr 110	. . . 4 |- ( ( (DECID ph /\ DECID ps ) /\ DECID ch ) -> DECID ch )
11		dcbi 938	. . . . . 6 |- (DECID ph -> (DECID ch -> DECID ( ph <-> ch ) ) )
12		dcbi 938	. . . . . 6 |- (DECID ps -> (DECID ( ph <-> ch ) -> DECID ( ps <-> ( ph <-> ch ) ) ) )
13	11, 12	syl9 72	. . . . 5 |- (DECID ph -> (DECID ps -> (DECID ch -> DECID ( ps <-> ( ph <-> ch ) ) ) ) )
14	13	imp31 256	. . . 4 |- ( ( (DECID ph /\ DECID ps ) /\ DECID ch ) -> DECID ( ps <-> ( ph <-> ch ) ) )
15		biassdc 1406	. . . 4 |- (DECID ( ph <-> ps ) -> (DECID ch -> (DECID ( ps <-> ( ph <-> ch ) ) -> ( ( ( ( ph <-> ps ) <-> ch ) <-> ( ps <-> ( ph <-> ch ) ) ) <-> ( ( ph <-> ps ) <-> ( ch <-> ( ps <-> ( ph <-> ch ) ) ) ) ) ) ) )
16	9, 10, 14, 15	syl3c 63	. . 3 |- ( ( (DECID ph /\ DECID ps ) /\ DECID ch ) -> ( ( ( ( ph <-> ps ) <-> ch ) <-> ( ps <-> ( ph <-> ch ) ) ) <-> ( ( ph <-> ps ) <-> ( ch <-> ( ps <-> ( ph <-> ch ) ) ) ) ) )
17	6, 16	mpbid 147	. 2 |- ( ( (DECID ph /\ DECID ps ) /\ DECID ch ) -> ( ( ph <-> ps ) <-> ( ch <-> ( ps <-> ( ph <-> ch ) ) ) ) )
18		simplr 528	. . 3 |- ( ( (DECID ph /\ DECID ps ) /\ DECID ch ) -> DECID ps )
19	11	imp 124	. . . 4 |- ( (DECID ph /\ DECID ch ) -> DECID ( ph <-> ch ) )
20	19	adantlr 477	. . 3 |- ( ( (DECID ph /\ DECID ps ) /\ DECID ch ) -> DECID ( ph <-> ch ) )
21		biassdc 1406	. . 3 |- (DECID ch -> (DECID ps -> (DECID ( ph <-> ch ) -> ( ( ( ch <-> ps ) <-> ( ph <-> ch ) ) <-> ( ch <-> ( ps <-> ( ph <-> ch ) ) ) ) ) ) )
22	10, 18, 20, 21	syl3c 63	. 2 |- ( ( (DECID ph /\ DECID ps ) /\ DECID ch ) -> ( ( ( ch <-> ps ) <-> ( ph <-> ch ) ) <-> ( ch <-> ( ps <-> ( ph <-> ch ) ) ) ) )
23	17, 22	bitr4d 191	1 |- ( ( (DECID ph /\ DECID ps ) /\ DECID ch ) -> ( ( ph <-> ps ) <-> ( ( ch <-> ps ) <-> ( ph <-> ch ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by: (None)