Theorem euotd 4009

Description: Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.)

Hypotheses

Ref	Expression
euotd.1	|- ( ph -> A e. _V )
euotd.2	|- ( ph -> B e. _V )
euotd.3	|- ( ph -> C e. _V )
euotd.4	|- ( ph -> ( ps <-> ( a = A /\ b = B /\ c = C ) ) )

Assertion

Ref	Expression
euotd	|- ( ph -> E! x E. a E. b E. c ( x = <. a , b , c >. /\ ps ) )

Distinct variable groups:   a, b, c, x, A   B, a, b, c, x   C, a, b, c, x   ph, a, b, c, x

Allowed substitution hints:   ps( x, a, b, c)

Proof of Theorem euotd

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euotd.1	. . . 4 |- ( ph -> A e. _V )
2		euotd.2	. . . 4 |- ( ph -> B e. _V )
3		euotd.3	. . . 4 |- ( ph -> C e. _V )
4		otexg 3985	. . . 4 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> <. A , B , C >. e. _V )
5	1, 2, 3, 4	syl3anc 1169	. . 3 |- ( ph -> <. A , B , C >. e. _V )
6		euotd.4	. . . . . . . . . . . . 13 |- ( ph -> ( ps <-> ( a = A /\ b = B /\ c = C ) ) )
7	6	biimpa 290	. . . . . . . . . . . 12 |- ( ( ph /\ ps ) -> ( a = A /\ b = B /\ c = C ) )
8		vex 2604	. . . . . . . . . . . . 13 |- a e. _V
9		vex 2604	. . . . . . . . . . . . 13 |- b e. _V
10		vex 2604	. . . . . . . . . . . . 13 |- c e. _V
11	8, 9, 10	otth 3997	. . . . . . . . . . . 12 |- ( <. a , b , c >. = <. A , B , C >. <-> ( a = A /\ b = B /\ c = C ) )
12	7, 11	sylibr 132	. . . . . . . . . . 11 |- ( ( ph /\ ps ) -> <. a , b , c >. = <. A , B , C >. )
13	12	eqeq2d 2092	. . . . . . . . . 10 |- ( ( ph /\ ps ) -> ( x = <. a , b , c >. <-> x = <. A , B , C >. ) )
14	13	biimpd 142	. . . . . . . . 9 |- ( ( ph /\ ps ) -> ( x = <. a , b , c >. -> x = <. A , B , C >. ) )
15	14	impancom 256	. . . . . . . 8 |- ( ( ph /\ x = <. a , b , c >. ) -> ( ps -> x = <. A , B , C >. ) )
16	15	expimpd 355	. . . . . . 7 |- ( ph -> ( ( x = <. a , b , c >. /\ ps ) -> x = <. A , B , C >. ) )
17	16	exlimdv 1740	. . . . . 6 |- ( ph -> ( E. c ( x = <. a , b , c >. /\ ps ) -> x = <. A , B , C >. ) )
18	17	exlimdvv 1818	. . . . 5 |- ( ph -> ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) -> x = <. A , B , C >. ) )
19		tru 1288	. . . . . . . . . . 11 |- T.
20	2	adantr 270	. . . . . . . . . . . . 13 |- ( ( ph /\ a = A ) -> B e. _V )
21	3	ad2antrr 471	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ a = A ) /\ b = B ) -> C e. _V )
22		simpr 108	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( a = A /\ b = B /\ c = C ) )
23	22, 11	sylibr 132	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> <. a , b , c >. = <. A , B , C >. )
24	23	eqcomd 2086	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> <. A , B , C >. = <. a , b , c >. )
25	6	biimpar 291	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ps )
26	24, 25	jca 300	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
27		a1tru 1300	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> T. )
28	26, 27	2thd 173	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
29	28	3anassrs 1160	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ a = A ) /\ b = B ) /\ c = C ) -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
30	21, 29	sbcied 2850	. . . . . . . . . . . . 13 |- ( ( ( ph /\ a = A ) /\ b = B ) -> ( [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
31	20, 30	sbcied 2850	. . . . . . . . . . . 12 |- ( ( ph /\ a = A ) -> ( [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
32	1, 31	sbcied 2850	. . . . . . . . . . 11 |- ( ph -> ( [. A / a ]. [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
33	19, 32	mpbiri 166	. . . . . . . . . 10 |- ( ph -> [. A / a ]. [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
34	33	spesbcd 2900	. . . . . . . . 9 |- ( ph -> E. a [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
35		nfcv 2219	. . . . . . . . . 10 |- F/_ b B
36		nfsbc1v 2833	. . . . . . . . . . 11 |- F/ b [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
37	36	nfex 1568	. . . . . . . . . 10 |- F/ b E. a [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
38		sbceq1a 2824	. . . . . . . . . . 11 |- ( b = B -> ( [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
39	38	exbidv 1746	. . . . . . . . . 10 |- ( b = B -> ( E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> E. a [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
40	35, 37, 39	spcegf 2681	. . . . . . . . 9 |- ( B e. _V -> ( E. a [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) -> E. b E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
41	2, 34, 40	sylc 61	. . . . . . . 8 |- ( ph -> E. b E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
42		nfcv 2219	. . . . . . . . 9 |- F/_ c C
43		nfsbc1v 2833	. . . . . . . . . . 11 |- F/ c [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
44	43	nfex 1568	. . . . . . . . . 10 |- F/ c E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
45	44	nfex 1568	. . . . . . . . 9 |- F/ c E. b E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
46		sbceq1a 2824	. . . . . . . . . 10 |- ( c = C -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
47	46	2exbidv 1789	. . . . . . . . 9 |- ( c = C -> ( E. b E. a ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> E. b E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
48	42, 45, 47	spcegf 2681	. . . . . . . 8 |- ( C e. _V -> ( E. b E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) -> E. c E. b E. a ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
49	3, 41, 48	sylc 61	. . . . . . 7 |- ( ph -> E. c E. b E. a ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
50		excom13 1619	. . . . . . 7 |- ( E. c E. b E. a ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> E. a E. b E. c ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
51	49, 50	sylib 120	. . . . . 6 |- ( ph -> E. a E. b E. c ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
52		eqeq1 2087	. . . . . . . 8 |- ( x = <. A , B , C >. -> ( x = <. a , b , c >. <-> <. A , B , C >. = <. a , b , c >. ) )
53	52	anbi1d 452	. . . . . . 7 |- ( x = <. A , B , C >. -> ( ( x = <. a , b , c >. /\ ps ) <-> ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
54	53	3exbidv 1790	. . . . . 6 |- ( x = <. A , B , C >. -> ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> E. a E. b E. c ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
55	51, 54	syl5ibrcom 155	. . . . 5 |- ( ph -> ( x = <. A , B , C >. -> E. a E. b E. c ( x = <. a , b , c >. /\ ps ) ) )
56	18, 55	impbid 127	. . . 4 |- ( ph -> ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = <. A , B , C >. ) )
57	56	alrimiv 1795	. . 3 |- ( ph -> A. x ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = <. A , B , C >. ) )
58		eqeq2 2090	. . . . . 6 |- ( y = <. A , B , C >. -> ( x = y <-> x = <. A , B , C >. ) )
59	58	bibi2d 230	. . . . 5 |- ( y = <. A , B , C >. -> ( ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = y ) <-> ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = <. A , B , C >. ) ) )
60	59	albidv 1745	. . . 4 |- ( y = <. A , B , C >. -> ( A. x ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = y ) <-> A. x ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = <. A , B , C >. ) ) )
61	60	spcegv 2686	. . 3 |- ( <. A , B , C >. e. _V -> ( A. x ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = <. A , B , C >. ) -> E. y A. x ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = y ) ) )
62	5, 57, 61	sylc 61	. 2 |- ( ph -> E. y A. x ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = y ) )
63		df-eu 1944	. 2 |- ( E! x E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> E. y A. x ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = y ) )
64	62, 63	sylibr 132	1 |- ( ph -> E! x E. a E. b E. c ( x = <. a , b , c >. /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   A.wal 1282   = wceq 1284   T. wtru 1285   E.wex 1421   e. wcel 1433   E!weu 1941   _Vcvv 2601   [.wsbc 2815   <.cotp 3402

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-ot 3408

This theorem is referenced by: (None)