Theorem eueq3dc 2709

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 842	. 2 |- (DECID ph -> (DECID ps -> DECID (ph \/ ps)))
2		df-dc 742	. . 3 |- (DECID (ph \/ ps) <-> ((ph \/ ps) \/ -. (ph \/ ps)))
3		eueq3dc.1	. . . . . . 7 |- A e. _V
4	3	eueq1 2707	. . . . . 6 |- E!x x = A
5		ibar 285	. . . . . . . . 9 |- (ph -> (x = A <-> (ph /\ x = A)))
6		pm2.45 656	. . . . . . . . . . . . 13 |- (-. (ph \/ ps) -> -. ph)
7		eueq3dc.4	. . . . . . . . . . . . . . 15 |- -. (ph /\ ps)
8	7	imnani 624	. . . . . . . . . . . . . 14 |- (ph -> -. ps)
9	8	con2i 557	. . . . . . . . . . . . 13 |- (ps -> -. ph)
10	6, 9	jaoi 635	. . . . . . . . . . . 12 |- ((-. (ph \/ ps) \/ ps) -> -. ph)
11	10	con2i 557	. . . . . . . . . . 11 |- (ph -> -. (-. (ph \/ ps) \/ ps))
12	6	con2i 557	. . . . . . . . . . . . 13 |- (ph -> -. -. (ph \/ ps))
13	12	bianfd 854	. . . . . . . . . . . 12 |- (ph -> (-. (ph \/ ps) <-> (-. (ph \/ ps) /\ x = B)))
14	8	bianfd 854	. . . . . . . . . . . 12 |- (ph -> (ps <-> (ps /\ x = C)))
15	13, 14	orbi12d 706	. . . . . . . . . . 11 |- (ph -> ((-. (ph \/ ps) \/ ps) <-> ((-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
16	11, 15	mtbid 596	. . . . . . . . . 10 |- (ph -> -. ((-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
17		biorf 662	. . . . . . . . . 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 177	. . . . . . . 8 |- (ph -> (x = A <-> (((-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) \/ (ph /\ x = A))))
20		3orrot 890	. . . . . . . . 9 |- (((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) <-> ((-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C) \/ (ph /\ x = A)))
21		df-3or 885	. . . . . . . . 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 173	. . . . . . . 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 187	. . . . . . 7 |- (ph -> (x = A <-> ((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
24	23	eubidv 1905	. . . . . 6 |- (ph -> (E!x x = A <-> E!x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
25	4, 24	mpbii 136	. . . . 5 |- (ph -> E!x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
26		eueq3dc.3	. . . . . . 7 |- C e. _V
27	26	eueq1 2707	. . . . . 6 |- E!x x = C
28		ibar 285	. . . . . . . . 9 |- (ps -> (x = C <-> (ps /\ x = C)))
29	8	adantr 261	. . . . . . . . . . . 12 |- ((ph /\ x = A) -> -. ps)
30		pm2.46 657	. . . . . . . . . . . . 13 |- (-. (ph \/ ps) -> -. ps)
31	30	adantr 261	. . . . . . . . . . . 12 |- ((-. (ph \/ ps) /\ x = B) -> -. ps)
32	29, 31	jaoi 635	. . . . . . . . . . 11 |- (((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B)) -> -. ps)
33	32	con2i 557	. . . . . . . . . 10 |- (ps -> -. ((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B)))
34		biorf 662	. . . . . . . . . 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 177	. . . . . . . 8 |- (ps -> (x = C <-> (((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B)) \/ (ps /\ x = C))))
37		df-3or 885	. . . . . . . 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 187	. . . . . . 7 |- (ps -> (x = C <-> ((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
39	38	eubidv 1905	. . . . . 6 |- (ps -> (E!x x = C <-> E!x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
40	27, 39	mpbii 136	. . . . 5 |- (ps -> E!x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
41	25, 40	jaoi 635	. . . 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 2707	. . . . 5 |- E!x x = B
44		ibar 285	. . . . . . . 8 |- (-. (ph \/ ps) -> (x = B <-> (-. (ph \/ ps) /\ x = B)))
45		simpl 102	. . . . . . . . . . 11 |- ((ph /\ x = A) -> ph)
46		simpl 102	. . . . . . . . . . 11 |- ((ps /\ x = C) -> ps)
47	45, 46	orim12i 675	. . . . . . . . . 10 |- (((ph /\ x = A) \/ (ps /\ x = C)) -> (ph \/ ps))
48	47	con3i 561	. . . . . . . . 9 |- (-. (ph \/ ps) -> -. ((ph /\ x = A) \/ (ps /\ x = C)))
49		biorf 662	. . . . . . . . 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 177	. . . . . . 7 |- (-. (ph \/ ps) -> (x = B <-> (((ph /\ x = A) \/ (ps /\ x = C)) \/ (-. (ph \/ ps) /\ x = B))))
52		3orcomb 893	. . . . . . . 8 |- (((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)) <-> ((ph /\ x = A) \/ (ps /\ x = C) \/ (-. (ph \/ ps) /\ x = B)))
53		df-3or 885	. . . . . . . 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 173	. . . . . . 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 187	. . . . . 6 |- (-. (ph \/ ps) -> (x = B <-> ((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
56	55	eubidv 1905	. . . . 5 |- (-. (ph \/ ps) -> (E!x x = B <-> E!x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))
57	43, 56	mpbii 136	. . . 4 |- (-. (ph \/ ps) -> E!x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
58	41, 57	jaoi 635	. . 3 |- (((ph \/ ps) \/ -. (ph \/ ps)) -> E!x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
59	2, 58	sylbi 114	. 2 |- (DECID (ph \/ ps) -> E!x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C)))
60	1, 59	syl6 29	1 |- (DECID ph -> (DECID ps -> E!x((ph /\ x = A) \/ (-. (ph \/ ps) /\ x = B) \/ (ps /\ x = C))))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 628  DECID wdc 741   \/ w3o 883   = wceq 1242   e. wcel 1390  E!weu 1897   _Vcvv 2551

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  moeq3dc  2711