Theorem euotd 4251

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 4227	. . . 4 |- ( ( A e. _V /\ B e. _V /\ C e. _V ) -> <. A , B , C >. e. _V )
5	1, 2, 3, 4	syl3anc 1238	. . 3 |- ( ph -> <. A , B , C >. e. _V )
6		euotd.4	. . . . . . . . . . . . 13 |- ( ph -> ( ps <-> ( a = A /\ b = B /\ c = C ) ) )
7	6	biimpa 296	. . . . . . . . . . . 12 |- ( ( ph /\ ps ) -> ( a = A /\ b = B /\ c = C ) )
8		vex 2740	. . . . . . . . . . . . 13 |- a e. _V
9		vex 2740	. . . . . . . . . . . . 13 |- b e. _V
10		vex 2740	. . . . . . . . . . . . 13 |- c e. _V
11	8, 9, 10	otth 4239	. . . . . . . . . . . 12 |- ( <. a , b , c >. = <. A , B , C >. <-> ( a = A /\ b = B /\ c = C ) )
12	7, 11	sylibr 134	. . . . . . . . . . 11 |- ( ( ph /\ ps ) -> <. a , b , c >. = <. A , B , C >. )
13	12	eqeq2d 2189	. . . . . . . . . 10 |- ( ( ph /\ ps ) -> ( x = <. a , b , c >. <-> x = <. A , B , C >. ) )
14	13	biimpd 144	. . . . . . . . 9 |- ( ( ph /\ ps ) -> ( x = <. a , b , c >. -> x = <. A , B , C >. ) )
15	14	impancom 260	. . . . . . . 8 |- ( ( ph /\ x = <. a , b , c >. ) -> ( ps -> x = <. A , B , C >. ) )
16	15	expimpd 363	. . . . . . 7 |- ( ph -> ( ( x = <. a , b , c >. /\ ps ) -> x = <. A , B , C >. ) )
17	16	exlimdv 1819	. . . . . 6 |- ( ph -> ( E. c ( x = <. a , b , c >. /\ ps ) -> x = <. A , B , C >. ) )
18	17	exlimdvv 1897	. . . . 5 |- ( ph -> ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) -> x = <. A , B , C >. ) )
19		tru 1357	. . . . . . . . . . 11 |- T.
20	2	adantr 276	. . . . . . . . . . . . 13 |- ( ( ph /\ a = A ) -> B e. _V )
21	3	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ a = A ) /\ b = B ) -> C e. _V )
22		simpr 110	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( a = A /\ b = B /\ c = C ) )
23	22, 11	sylibr 134	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> <. a , b , c >. = <. A , B , C >. )
24	23	eqcomd 2183	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> <. A , B , C >. = <. a , b , c >. )
25	6	biimpar 297	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ps )
26	24, 25	jca 306	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
27		a1tru 1369	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> T. )
28	26, 27	2thd 175	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
29	28	3anassrs 1229	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ a = A ) /\ b = B ) /\ c = C ) -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
30	21, 29	sbcied 2999	. . . . . . . . . . . . 13 |- ( ( ( ph /\ a = A ) /\ b = B ) -> ( [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
31	20, 30	sbcied 2999	. . . . . . . . . . . 12 |- ( ( ph /\ a = A ) -> ( [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
32	1, 31	sbcied 2999	. . . . . . . . . . 11 |- ( ph -> ( [. A / a ]. [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
33	19, 32	mpbiri 168	. . . . . . . . . 10 |- ( ph -> [. A / a ]. [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
34	33	spesbcd 3049	. . . . . . . . 9 |- ( ph -> E. a [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
35		nfcv 2319	. . . . . . . . . 10 |- F/_ b B
36		nfsbc1v 2981	. . . . . . . . . . 11 |- F/ b [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
37	36	nfex 1637	. . . . . . . . . 10 |- F/ b E. a [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
38		sbceq1a 2972	. . . . . . . . . . 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 1825	. . . . . . . . . 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 2820	. . . . . . . . 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 2319	. . . . . . . . 9 |- F/_ c C
43		nfsbc1v 2981	. . . . . . . . . . 11 |- F/ c [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
44	43	nfex 1637	. . . . . . . . . 10 |- F/ c E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
45	44	nfex 1637	. . . . . . . . 9 |- F/ c E. b E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
46		sbceq1a 2972	. . . . . . . . . 10 |- ( c = C -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
47	46	2exbidv 1868	. . . . . . . . 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 2820	. . . . . . . 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 1689	. . . . . . 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 122	. . . . . 6 |- ( ph -> E. a E. b E. c ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
52		eqeq1 2184	. . . . . . . 8 |- ( x = <. A , B , C >. -> ( x = <. a , b , c >. <-> <. A , B , C >. = <. a , b , c >. ) )
53	52	anbi1d 465	. . . . . . 7 |- ( x = <. A , B , C >. -> ( ( x = <. a , b , c >. /\ ps ) <-> ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
54	53	3exbidv 1869	. . . . . 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 157	. . . . 5 |- ( ph -> ( x = <. A , B , C >. -> E. a E. b E. c ( x = <. a , b , c >. /\ ps ) ) )
56	18, 55	impbid 129	. . . 4 |- ( ph -> ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = <. A , B , C >. ) )
57	56	alrimiv 1874	. . 3 |- ( ph -> A. x ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = <. A , B , C >. ) )
58		eqeq2 2187	. . . . . 6 |- ( y = <. A , B , C >. -> ( x = y <-> x = <. A , B , C >. ) )
59	58	bibi2d 232	. . . . 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 1824	. . . 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 2825	. . 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 2029	. 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 134	1 |- ( ph -> E! x E. a E. b E. c ( x = <. a , b , c >. /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   A.wal 1351   = wceq 1353   T. wtru 1354   E.wex 1492   E!weu 2026   e. wcel 2148   _Vcvv 2737   [.wsbc 2962   <.cotp 3595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-ot 3601

This theorem is referenced by: (None)