Theorem euotd 4114

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 4090	. . . 4 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> <. A , B , C >. e. _V )
5	1, 2, 3, 4	syl3anc 1184	. . 3 |- ( ph -> <. A , B , C >. e. _V )
6		euotd.4	. . . . . . . . . . . . 13 |- ( ph -> ( ps <-> ( a = A /\ b = B /\ c = C ) ) )
7	6	biimpa 292	. . . . . . . . . . . 12 |- ( ( ph /\ ps ) -> ( a = A /\ b = B /\ c = C ) )
8		vex 2644	. . . . . . . . . . . . 13 |- a e. _V
9		vex 2644	. . . . . . . . . . . . 13 |- b e. _V
10		vex 2644	. . . . . . . . . . . . 13 |- c e. _V
11	8, 9, 10	otth 4102	. . . . . . . . . . . 12 |- ( <. a , b , c >. = <. A , B , C >. <-> ( a = A /\ b = B /\ c = C ) )
12	7, 11	sylibr 133	. . . . . . . . . . 11 |- ( ( ph /\ ps ) -> <. a , b , c >. = <. A , B , C >. )
13	12	eqeq2d 2111	. . . . . . . . . 10 |- ( ( ph /\ ps ) -> ( x = <. a , b , c >. <-> x = <. A , B , C >. ) )
14	13	biimpd 143	. . . . . . . . 9 |- ( ( ph /\ ps ) -> ( x = <. a , b , c >. -> x = <. A , B , C >. ) )
15	14	impancom 258	. . . . . . . 8 |- ( ( ph /\ x = <. a , b , c >. ) -> ( ps -> x = <. A , B , C >. ) )
16	15	expimpd 358	. . . . . . 7 |- ( ph -> ( ( x = <. a , b , c >. /\ ps ) -> x = <. A , B , C >. ) )
17	16	exlimdv 1758	. . . . . 6 |- ( ph -> ( E. c ( x = <. a , b , c >. /\ ps ) -> x = <. A , B , C >. ) )
18	17	exlimdvv 1836	. . . . 5 |- ( ph -> ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) -> x = <. A , B , C >. ) )
19		tru 1303	. . . . . . . . . . 11 |- T.
20	2	adantr 272	. . . . . . . . . . . . 13 |- ( ( ph /\ a = A ) -> B e. _V )
21	3	ad2antrr 475	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ a = A ) /\ b = B ) -> C e. _V )
22		simpr 109	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( a = A /\ b = B /\ c = C ) )
23	22, 11	sylibr 133	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> <. a , b , c >. = <. A , B , C >. )
24	23	eqcomd 2105	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> <. A , B , C >. = <. a , b , c >. )
25	6	biimpar 293	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ps )
26	24, 25	jca 302	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
27		a1tru 1315	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> T. )
28	26, 27	2thd 174	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
29	28	3anassrs 1175	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ a = A ) /\ b = B ) /\ c = C ) -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
30	21, 29	sbcied 2897	. . . . . . . . . . . . 13 |- ( ( ( ph /\ a = A ) /\ b = B ) -> ( [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
31	20, 30	sbcied 2897	. . . . . . . . . . . 12 |- ( ( ph /\ a = A ) -> ( [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
32	1, 31	sbcied 2897	. . . . . . . . . . 11 |- ( ph -> ( [. A / a ]. [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
33	19, 32	mpbiri 167	. . . . . . . . . 10 |- ( ph -> [. A / a ]. [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
34	33	spesbcd 2947	. . . . . . . . 9 |- ( ph -> E. a [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
35		nfcv 2240	. . . . . . . . . 10 |- F/_ b B
36		nfsbc1v 2880	. . . . . . . . . . 11 |- F/ b [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
37	36	nfex 1584	. . . . . . . . . 10 |- F/ b E. a [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
38		sbceq1a 2871	. . . . . . . . . . 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 1764	. . . . . . . . . 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 2724	. . . . . . . . 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 62	. . . . . . . 8 |- ( ph -> E. b E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
42		nfcv 2240	. . . . . . . . 9 |- F/_ c C
43		nfsbc1v 2880	. . . . . . . . . . 11 |- F/ c [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
44	43	nfex 1584	. . . . . . . . . 10 |- F/ c E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
45	44	nfex 1584	. . . . . . . . 9 |- F/ c E. b E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
46		sbceq1a 2871	. . . . . . . . . 10 |- ( c = C -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
47	46	2exbidv 1807	. . . . . . . . 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 2724	. . . . . . . 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 62	. . . . . . 7 |- ( ph -> E. c E. b E. a ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
50		excom13 1635	. . . . . . 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 121	. . . . . 6 |- ( ph -> E. a E. b E. c ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
52		eqeq1 2106	. . . . . . . 8 |- ( x = <. A , B , C >. -> ( x = <. a , b , c >. <-> <. A , B , C >. = <. a , b , c >. ) )
53	52	anbi1d 456	. . . . . . 7 |- ( x = <. A , B , C >. -> ( ( x = <. a , b , c >. /\ ps ) <-> ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
54	53	3exbidv 1808	. . . . . 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 156	. . . . 5 |- ( ph -> ( x = <. A , B , C >. -> E. a E. b E. c ( x = <. a , b , c >. /\ ps ) ) )
56	18, 55	impbid 128	. . . 4 |- ( ph -> ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = <. A , B , C >. ) )
57	56	alrimiv 1813	. . 3 |- ( ph -> A. x ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = <. A , B , C >. ) )
58		eqeq2 2109	. . . . . 6 |- ( y = <. A , B , C >. -> ( x = y <-> x = <. A , B , C >. ) )
59	58	bibi2d 231	. . . . 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 1763	. . . 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 2729	. . 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 62	. 2 |- ( ph -> E. y A. x ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = y ) )
63		df-eu 1963	. 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 133	1 |- ( ph -> E! x E. a E. b E. c ( x = <. a , b , c >. /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 930   A.wal 1297   = wceq 1299   T. wtru 1300   E.wex 1436   e. wcel 1448   E!weu 1960   _Vcvv 2641   [.wsbc 2862   <.cotp 3478

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-ot 3484

This theorem is referenced by: (None)