Theorem euotd 4741

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.4	. . . . . . . . . . . . 13 |- ( ph -> ( ps <-> ( a = A /\ b = B /\ c = C ) ) )
2	1	biimpa 484	. . . . . . . . . . . 12 |- ( ( ph /\ ps ) -> ( a = A /\ b = B /\ c = C ) )
3		vex 3109	. . . . . . . . . . . . 13 |- a e. _V
4		vex 3109	. . . . . . . . . . . . 13 |- b e. _V
5		vex 3109	. . . . . . . . . . . . 13 |- c e. _V
6	3, 4, 5	otth 4722	. . . . . . . . . . . 12 |- ( <. a , b , c >. = <. A , B , C >. <-> ( a = A /\ b = B /\ c = C ) )
7	2, 6	sylibr 212	. . . . . . . . . . 11 |- ( ( ph /\ ps ) -> <. a , b , c >. = <. A , B , C >. )
8	7	eqeq2d 2474	. . . . . . . . . 10 |- ( ( ph /\ ps ) -> ( x = <. a , b , c >. <-> x = <. A , B , C >. ) )
9	8	biimpd 207	. . . . . . . . 9 |- ( ( ph /\ ps ) -> ( x = <. a , b , c >. -> x = <. A , B , C >. ) )
10	9	impancom 440	. . . . . . . 8 |- ( ( ph /\ x = <. a , b , c >. ) -> ( ps -> x = <. A , B , C >. ) )
11	10	expimpd 603	. . . . . . 7 |- ( ph -> ( ( x = <. a , b , c >. /\ ps ) -> x = <. A , B , C >. ) )
12	11	exlimdv 1695	. . . . . 6 |- ( ph -> ( E. c ( x = <. a , b , c >. /\ ps ) -> x = <. A , B , C >. ) )
13	12	exlimdvv 1696	. . . . 5 |- ( ph -> ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) -> x = <. A , B , C >. ) )
14		euotd.3	. . . . . . . 8 |- ( ph -> C e. _V )
15		euotd.2	. . . . . . . . 9 |- ( ph -> B e. _V )
16		tru 1378	. . . . . . . . . . 11 |- T.
17		euotd.1	. . . . . . . . . . . 12 |- ( ph -> A e. _V )
18	15	adantr 465	. . . . . . . . . . . . 13 |- ( ( ph /\ a = A ) -> B e. _V )
19	14	ad2antrr 725	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ a = A ) /\ b = B ) -> C e. _V )
20		simpr 461	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( a = A /\ b = B /\ c = C ) )
21	20, 6	sylibr 212	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> <. a , b , c >. = <. A , B , C >. )
22	21	eqcomd 2468	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> <. A , B , C >. = <. a , b , c >. )
23	1	biimpar 485	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ps )
24	22, 23	jca 532	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
25		a1tru 1390	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> T. )
26	24, 25	2thd 240	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
27	26	3anassrs 1213	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ a = A ) /\ b = B ) /\ c = C ) -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
28	19, 27	sbcied 3361	. . . . . . . . . . . . 13 |- ( ( ( ph /\ a = A ) /\ b = B ) -> ( [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
29	18, 28	sbcied 3361	. . . . . . . . . . . 12 |- ( ( ph /\ a = A ) -> ( [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
30	17, 29	sbcied 3361	. . . . . . . . . . 11 |- ( ph -> ( [. A / a ]. [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
31	16, 30	mpbiri 233	. . . . . . . . . 10 |- ( ph -> [. A / a ]. [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
32	31	spesbcd 3418	. . . . . . . . 9 |- ( ph -> E. a [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
33		nfcv 2622	. . . . . . . . . 10 |- F/_ b B
34		nfsbc1v 3344	. . . . . . . . . . 11 |- F/ b [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
35	34	nfex 1890	. . . . . . . . . 10 |- F/ b E. a [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
36		sbceq1a 3335	. . . . . . . . . . 11 |- ( b = B -> ( [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
37	36	exbidv 1685	. . . . . . . . . 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 ) ) )
38	33, 35, 37	spcegf 3187	. . . . . . . . 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 ) ) )
39	15, 32, 38	sylc 60	. . . . . . . 8 |- ( ph -> E. b E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
40		nfcv 2622	. . . . . . . . 9 |- F/_ c C
41		nfsbc1v 3344	. . . . . . . . . . 11 |- F/ c [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
42	41	nfex 1890	. . . . . . . . . 10 |- F/ c E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
43	42	nfex 1890	. . . . . . . . 9 |- F/ c E. b E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
44		sbceq1a 3335	. . . . . . . . . 10 |- ( c = C -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
45	44	2exbidv 1687	. . . . . . . . 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 ) ) )
46	40, 43, 45	spcegf 3187	. . . . . . . 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 ) ) )
47	14, 39, 46	sylc 60	. . . . . . 7 |- ( ph -> E. c E. b E. a ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
48		excom13 1795	. . . . . . 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 ) )
49	47, 48	sylib 196	. . . . . 6 |- ( ph -> E. a E. b E. c ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
50		eqeq1 2464	. . . . . . . 8 |- ( x = <. A , B , C >. -> ( x = <. a , b , c >. <-> <. A , B , C >. = <. a , b , c >. ) )
51	50	anbi1d 704	. . . . . . 7 |- ( x = <. A , B , C >. -> ( ( x = <. a , b , c >. /\ ps ) <-> ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
52	51	3exbidv 1688	. . . . . 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 ) ) )
53	49, 52	syl5ibrcom 222	. . . . 5 |- ( ph -> ( x = <. A , B , C >. -> E. a E. b E. c ( x = <. a , b , c >. /\ ps ) ) )
54	13, 53	impbid 191	. . . 4 |- ( ph -> ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = <. A , B , C >. ) )
55	54	alrimiv 1690	. . 3 |- ( ph -> A. x ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = <. A , B , C >. ) )
56		otex 4705	. . . 4 |- <. A , B , C >. e. _V
57		eqeq2 2475	. . . . . 6 |- ( y = <. A , B , C >. -> ( x = y <-> x = <. A , B , C >. ) )
58	57	bibi2d 318	. . . . 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 >. ) ) )
59	58	albidv 1684	. . . 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 >. ) ) )
60	56, 59	spcev 3198	. . 3 |- ( 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 ) )
61	55, 60	syl 16	. 2 |- ( ph -> E. y A. x ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = y ) )
62		df-eu 2272	. 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 ) )
63	61, 62	sylibr 212	1 |- ( ph -> E! x E. a E. b E. c ( x = <. a , b , c >. /\ ps ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 968   A.wal 1372   = wceq 1374   T. wtru 1375   E.wex 1591   e. wcel 1762   E!weu 2268   _Vcvv 3106   [.wsbc 3324   <.cotp 4028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-ot 4029

This theorem is referenced by:  oeeu  7242