Theorem biassdc 1286

Description: Associative law for the biconditional, for decidable propositions.

The classical version (without the decidability conditions) is an axiom of system DS in Vladimir Lifschitz, "On calculational proofs", Annals of Pure and Applied Logic, 113:207-224, 2002, http://www.cs.utexas.edu/users/ai-lab/pub-view.php?PubID=26805, and, interestingly, was not included in Principia Mathematica but was apparently first noted by Jan Lukasiewicz circa 1923. (Contributed by Jim Kingdon, 4-May-2018.)

Assertion

Ref	Expression
biassdc	|- (DECID ph -> (DECID ps -> (DECID ch -> ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) ) ) )

Proof of Theorem biassdc

Step	Hyp	Ref	Expression
1		df-dc 743	. . 3 |- (DECID ph <-> ( ph \/ -. ph ) )
2		pm5.501 233	. . . . . . 7 |- ( ph -> ( ps <-> ( ph <-> ps ) ) )
3	2	bibi1d 222	. . . . . 6 |- ( ph -> ( ( ps <-> ch ) <-> ( ( ph <-> ps ) <-> ch ) ) )
4		pm5.501 233	. . . . . 6 |- ( ph -> ( ( ps <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) )
5	3, 4	bitr3d 179	. . . . 5 |- ( ph -> ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) )
6	5	a1d 22	. . . 4 |- ( ph -> ( (DECID ps /\ DECID ch ) -> ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) ) )
7		nbbndc 1285	. . . . . . . . 9 |- (DECID ps -> (DECID ch -> ( ( -. ps <-> ch ) <-> -. ( ps <-> ch ) ) ) )
8	7	imp 115	. . . . . . . 8 |- ( (DECID ps /\ DECID ch ) -> ( ( -. ps <-> ch ) <-> -. ( ps <-> ch ) ) )
9	8	adantl 262	. . . . . . 7 |- ( ( -. ph /\ (DECID ps /\ DECID ch ) ) -> ( ( -. ps <-> ch ) <-> -. ( ps <-> ch ) ) )
10		nbn2 613	. . . . . . . . 9 |- ( -. ph -> ( -. ps <-> ( ph <-> ps ) ) )
11	10	bibi1d 222	. . . . . . . 8 |- ( -. ph -> ( ( -. ps <-> ch ) <-> ( ( ph <-> ps ) <-> ch ) ) )
12	11	adantr 261	. . . . . . 7 |- ( ( -. ph /\ (DECID ps /\ DECID ch ) ) -> ( ( -. ps <-> ch ) <-> ( ( ph <-> ps ) <-> ch ) ) )
13	9, 12	bitr3d 179	. . . . . 6 |- ( ( -. ph /\ (DECID ps /\ DECID ch ) ) -> ( -. ( ps <-> ch ) <-> ( ( ph <-> ps ) <-> ch ) ) )
14		nbn2 613	. . . . . . 7 |- ( -. ph -> ( -. ( ps <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) )
15	14	adantr 261	. . . . . 6 |- ( ( -. ph /\ (DECID ps /\ DECID ch ) ) -> ( -. ( ps <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) )
16	13, 15	bitr3d 179	. . . . 5 |- ( ( -. ph /\ (DECID ps /\ DECID ch ) ) -> ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) )
17	16	ex 108	. . . 4 |- ( -. ph -> ( (DECID ps /\ DECID ch ) -> ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) ) )
18	6, 17	jaoi 636	. . 3 |- ( ( ph \/ -. ph ) -> ( (DECID ps /\ DECID ch ) -> ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) ) )
19	1, 18	sylbi 114	. 2 |- (DECID ph -> ( (DECID ps /\ DECID ch ) -> ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) ) )
20	19	expd 245	1 |- (DECID ph -> (DECID ps -> (DECID ch -> ( ( ( ph <-> ps ) <-> ch ) <-> ( ph <-> ( ps <-> ch ) ) ) ) ) )

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:  bilukdc  1287