Theorem or3or 38319

Description: Decompose disjunction into three cases. (Contributed by RP, 5-Jul-2021.)

Assertion

Ref	Expression
or3or	|- ( ( ph \/ ps ) <-> ( ( ph /\ ps ) \/ ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) )

Proof of Theorem or3or

Step	Hyp	Ref	Expression
1		excxor 1469	. . 3 |- ( ( ph \/_ ps ) <-> ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) )
2	1	orbi2i 541	. 2 |- ( ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) <-> ( ( ph /\ ps ) \/ ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) ) )
3		orc 400	. . . 4 |- ( ph -> ( ph \/ ps ) )
4		exmid 431	. . . . 5 |- ( ps \/ -. ps )
5		pm3.2 463	. . . . . 6 |- ( ph -> ( ps -> ( ph /\ ps ) ) )
6		biimp 205	. . . . . . . . . 10 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
7		iman 440	. . . . . . . . . 10 |- ( ( ph -> ps ) <-> -. ( ph /\ -. ps ) )
8	6, 7	sylib 208	. . . . . . . . 9 |- ( ( ph <-> ps ) -> -. ( ph /\ -. ps ) )
9	8	con2i 134	. . . . . . . 8 |- ( ( ph /\ -. ps ) -> -. ( ph <-> ps ) )
10	9	ex 450	. . . . . . 7 |- ( ph -> ( -. ps -> -. ( ph <-> ps ) ) )
11		df-xor 1465	. . . . . . . 8 |- ( ( ph \/_ ps ) <-> -. ( ph <-> ps ) )
12	11	bicomi 214	. . . . . . 7 |- ( -. ( ph <-> ps ) <-> ( ph \/_ ps ) )
13	10, 12	syl6ib 241	. . . . . 6 |- ( ph -> ( -. ps -> ( ph \/_ ps ) ) )
14	5, 13	orim12d 883	. . . . 5 |- ( ph -> ( ( ps \/ -. ps ) -> ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) ) )
15	4, 14	mpi 20	. . . 4 |- ( ph -> ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) )
16	3, 15	2thd 255	. . 3 |- ( ph -> ( ( ph \/ ps ) <-> ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) ) )
17		bicom 212	. . . . . . 7 |- ( ( ph <-> ps ) <-> ( ps <-> ph ) )
18		bibif 361	. . . . . . 7 |- ( -. ph -> ( ( ps <-> ph ) <-> -. ps ) )
19	17, 18	syl5bb 272	. . . . . 6 |- ( -. ph -> ( ( ph <-> ps ) <-> -. ps ) )
20	19	con2bid 344	. . . . 5 |- ( -. ph -> ( ps <-> -. ( ph <-> ps ) ) )
21	20, 12	syl6bb 276	. . . 4 |- ( -. ph -> ( ps <-> ( ph \/_ ps ) ) )
22		biorf 420	. . . 4 |- ( -. ph -> ( ps <-> ( ph \/ ps ) ) )
23		simpl 473	. . . . . 6 |- ( ( ph /\ ps ) -> ph )
24	23	con3i 150	. . . . 5 |- ( -. ph -> -. ( ph /\ ps ) )
25		biorf 420	. . . . 5 |- ( -. ( ph /\ ps ) -> ( ( ph \/_ ps ) <-> ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) ) )
26	24, 25	syl 17	. . . 4 |- ( -. ph -> ( ( ph \/_ ps ) <-> ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) ) )
27	21, 22, 26	3bitr3d 298	. . 3 |- ( -. ph -> ( ( ph \/ ps ) <-> ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) ) )
28	16, 27	pm2.61i 176	. 2 |- ( ( ph \/ ps ) <-> ( ( ph /\ ps ) \/ ( ph \/_ ps ) ) )
29		3orass 1040	. 2 |- ( ( ( ph /\ ps ) \/ ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) <-> ( ( ph /\ ps ) \/ ( ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) ) )
30	2, 28, 29	3bitr4i 292	1 |- ( ( ph \/ ps ) <-> ( ( ph /\ ps ) \/ ( ph /\ -. ps ) \/ ( -. ph /\ ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   \/_ wxo 1464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-xor 1465

This theorem is referenced by:  uneqsn  38321