Theorem eueq3dc 2858

Description: Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.)

Hypotheses

Ref	Expression
eueq3dc.1	|- A e. _V
eueq3dc.2	|- B e. _V
eueq3dc.3	|- C e. _V
eueq3dc.4	|- -. ( ph /\ ps )

Assertion

Ref	Expression
eueq3dc	|- (DECID ph -> (DECID ps -> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )

Distinct variable groups:   ph, x   ps, x   x, A   x, B   x, C

Proof of Theorem eueq3dc

Step	Hyp	Ref	Expression
1		dcor 919	. 2 |- (DECID ph -> (DECID ps -> DECID ( ph \/ ps ) ) )
2		df-dc 820	. . 3 |- (DECID ( ph \/ ps ) <-> ( ( ph \/ ps ) \/ -. ( ph \/ ps ) ) )
3		eueq3dc.1	. . . . . . 7 |- A e. _V
4	3	eueq1 2856	. . . . . 6 |- E! x x = A
5		ibar 299	. . . . . . . . 9 |- ( ph -> ( x = A <-> ( ph /\ x = A ) ) )
6		pm2.45 727	. . . . . . . . . . . . 13 |- ( -. ( ph \/ ps ) -> -. ph )
7		eueq3dc.4	. . . . . . . . . . . . . . 15 |- -. ( ph /\ ps )
8	7	imnani 680	. . . . . . . . . . . . . 14 |- ( ph -> -. ps )
9	8	con2i 616	. . . . . . . . . . . . 13 |- ( ps -> -. ph )
10	6, 9	jaoi 705	. . . . . . . . . . . 12 |- ( ( -. ( ph \/ ps ) \/ ps ) -> -. ph )
11	10	con2i 616	. . . . . . . . . . 11 |- ( ph -> -. ( -. ( ph \/ ps ) \/ ps ) )
12	6	con2i 616	. . . . . . . . . . . . 13 |- ( ph -> -. -. ( ph \/ ps ) )
13	12	bianfd 932	. . . . . . . . . . . 12 |- ( ph -> ( -. ( ph \/ ps ) <-> ( -. ( ph \/ ps ) /\ x = B ) ) )
14	8	bianfd 932	. . . . . . . . . . . 12 |- ( ph -> ( ps <-> ( ps /\ x = C ) ) )
15	13, 14	orbi12d 782	. . . . . . . . . . 11 |- ( ph -> ( ( -. ( ph \/ ps ) \/ ps ) <-> ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
16	11, 15	mtbid 661	. . . . . . . . . 10 |- ( ph -> -. ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) )
17		biorf 733	. . . . . . . . . 10 |- ( -. ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) -> ( ( ph /\ x = A ) <-> ( ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) \/ ( ph /\ x = A ) ) ) )
18	16, 17	syl 14	. . . . . . . . 9 |- ( ph -> ( ( ph /\ x = A ) <-> ( ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) \/ ( ph /\ x = A ) ) ) )
19	5, 18	bitrd 187	. . . . . . . 8 |- ( ph -> ( x = A <-> ( ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) \/ ( ph /\ x = A ) ) ) )
20		3orrot 968	. . . . . . . . 9 |- ( ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) <-> ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) \/ ( ph /\ x = A ) ) )
21		df-3or 963	. . . . . . . . 9 |- ( ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) \/ ( ph /\ x = A ) ) <-> ( ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) \/ ( ph /\ x = A ) ) )
22	20, 21	bitri 183	. . . . . . . 8 |- ( ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) <-> ( ( ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) \/ ( ph /\ x = A ) ) )
23	19, 22	syl6bbr 197	. . . . . . 7 |- ( ph -> ( x = A <-> ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
24	23	eubidv 2007	. . . . . 6 |- ( ph -> ( E! x x = A <-> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
25	4, 24	mpbii 147	. . . . 5 |- ( ph -> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) )
26		eueq3dc.3	. . . . . . 7 |- C e. _V
27	26	eueq1 2856	. . . . . 6 |- E! x x = C
28		ibar 299	. . . . . . . . 9 |- ( ps -> ( x = C <-> ( ps /\ x = C ) ) )
29	8	adantr 274	. . . . . . . . . . . 12 |- ( ( ph /\ x = A ) -> -. ps )
30		pm2.46 728	. . . . . . . . . . . . 13 |- ( -. ( ph \/ ps ) -> -. ps )
31	30	adantr 274	. . . . . . . . . . . 12 |- ( ( -. ( ph \/ ps ) /\ x = B ) -> -. ps )
32	29, 31	jaoi 705	. . . . . . . . . . 11 |- ( ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) ) -> -. ps )
33	32	con2i 616	. . . . . . . . . 10 |- ( ps -> -. ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) ) )
34		biorf 733	. . . . . . . . . 10 |- ( -. ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) ) -> ( ( ps /\ x = C ) <-> ( ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) ) \/ ( ps /\ x = C ) ) ) )
35	33, 34	syl 14	. . . . . . . . 9 |- ( ps -> ( ( ps /\ x = C ) <-> ( ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) ) \/ ( ps /\ x = C ) ) ) )
36	28, 35	bitrd 187	. . . . . . . 8 |- ( ps -> ( x = C <-> ( ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) ) \/ ( ps /\ x = C ) ) ) )
37		df-3or 963	. . . . . . . 8 |- ( ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) <-> ( ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) ) \/ ( ps /\ x = C ) ) )
38	36, 37	syl6bbr 197	. . . . . . 7 |- ( ps -> ( x = C <-> ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
39	38	eubidv 2007	. . . . . 6 |- ( ps -> ( E! x x = C <-> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
40	27, 39	mpbii 147	. . . . 5 |- ( ps -> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) )
41	25, 40	jaoi 705	. . . 4 |- ( ( ph \/ ps ) -> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) )
42		eueq3dc.2	. . . . . 6 |- B e. _V
43	42	eueq1 2856	. . . . 5 |- E! x x = B
44		ibar 299	. . . . . . . 8 |- ( -. ( ph \/ ps ) -> ( x = B <-> ( -. ( ph \/ ps ) /\ x = B ) ) )
45		simpl 108	. . . . . . . . . . 11 |- ( ( ph /\ x = A ) -> ph )
46		simpl 108	. . . . . . . . . . 11 |- ( ( ps /\ x = C ) -> ps )
47	45, 46	orim12i 748	. . . . . . . . . 10 |- ( ( ( ph /\ x = A ) \/ ( ps /\ x = C ) ) -> ( ph \/ ps ) )
48	47	con3i 621	. . . . . . . . 9 |- ( -. ( ph \/ ps ) -> -. ( ( ph /\ x = A ) \/ ( ps /\ x = C ) ) )
49		biorf 733	. . . . . . . . 9 |- ( -. ( ( ph /\ x = A ) \/ ( ps /\ x = C ) ) -> ( ( -. ( ph \/ ps ) /\ x = B ) <-> ( ( ( ph /\ x = A ) \/ ( ps /\ x = C ) ) \/ ( -. ( ph \/ ps ) /\ x = B ) ) ) )
50	48, 49	syl 14	. . . . . . . 8 |- ( -. ( ph \/ ps ) -> ( ( -. ( ph \/ ps ) /\ x = B ) <-> ( ( ( ph /\ x = A ) \/ ( ps /\ x = C ) ) \/ ( -. ( ph \/ ps ) /\ x = B ) ) ) )
51	44, 50	bitrd 187	. . . . . . 7 |- ( -. ( ph \/ ps ) -> ( x = B <-> ( ( ( ph /\ x = A ) \/ ( ps /\ x = C ) ) \/ ( -. ( ph \/ ps ) /\ x = B ) ) ) )
52		3orcomb 971	. . . . . . . 8 |- ( ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) <-> ( ( ph /\ x = A ) \/ ( ps /\ x = C ) \/ ( -. ( ph \/ ps ) /\ x = B ) ) )
53		df-3or 963	. . . . . . . 8 |- ( ( ( ph /\ x = A ) \/ ( ps /\ x = C ) \/ ( -. ( ph \/ ps ) /\ x = B ) ) <-> ( ( ( ph /\ x = A ) \/ ( ps /\ x = C ) ) \/ ( -. ( ph \/ ps ) /\ x = B ) ) )
54	52, 53	bitri 183	. . . . . . 7 |- ( ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) <-> ( ( ( ph /\ x = A ) \/ ( ps /\ x = C ) ) \/ ( -. ( ph \/ ps ) /\ x = B ) ) )
55	51, 54	syl6bbr 197	. . . . . 6 |- ( -. ( ph \/ ps ) -> ( x = B <-> ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
56	55	eubidv 2007	. . . . 5 |- ( -. ( ph \/ ps ) -> ( E! x x = B <-> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
57	43, 56	mpbii 147	. . . 4 |- ( -. ( ph \/ ps ) -> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) )
58	41, 57	jaoi 705	. . 3 |- ( ( ( ph \/ ps ) \/ -. ( ph \/ ps ) ) -> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) )
59	2, 58	sylbi 120	. 2 |- (DECID ( ph \/ ps ) -> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) )
60	1, 59	syl6 33	1 |- (DECID ph -> (DECID ps -> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697  DECID wdc 819   \/ w3o 961   = wceq 1331   e. wcel 1480   E!weu 1999   _Vcvv 2686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688

This theorem is referenced by:  moeq3dc  2860