Theorem euotd 4722

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 486	. . . . . . . . . . . 12 |- ( ( ph /\ ps ) -> ( a = A /\ b = B /\ c = C ) )
3		vex 3090	. . . . . . . . . . . . 13 |- a e. _V
4		vex 3090	. . . . . . . . . . . . 13 |- b e. _V
5		vex 3090	. . . . . . . . . . . . 13 |- c e. _V
6	3, 4, 5	otth 4704	. . . . . . . . . . . 12 |- ( <. a , b , c >. = <. A , B , C >. <-> ( a = A /\ b = B /\ c = C ) )
7	2, 6	sylibr 215	. . . . . . . . . . 11 |- ( ( ph /\ ps ) -> <. a , b , c >. = <. A , B , C >. )
8	7	eqeq2d 2443	. . . . . . . . . 10 |- ( ( ph /\ ps ) -> ( x = <. a , b , c >. <-> x = <. A , B , C >. ) )
9	8	biimpd 210	. . . . . . . . 9 |- ( ( ph /\ ps ) -> ( x = <. a , b , c >. -> x = <. A , B , C >. ) )
10	9	impancom 441	. . . . . . . 8 |- ( ( ph /\ x = <. a , b , c >. ) -> ( ps -> x = <. A , B , C >. ) )
11	10	expimpd 606	. . . . . . 7 |- ( ph -> ( ( x = <. a , b , c >. /\ ps ) -> x = <. A , B , C >. ) )
12	11	exlimdv 1771	. . . . . 6 |- ( ph -> ( E. c ( x = <. a , b , c >. /\ ps ) -> x = <. A , B , C >. ) )
13	12	exlimdvv 1772	. . . . 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 1441	. . . . . . . . . . 11 |- T.
17		euotd.1	. . . . . . . . . . . 12 |- ( ph -> A e. _V )
18	15	adantr 466	. . . . . . . . . . . . 13 |- ( ( ph /\ a = A ) -> B e. _V )
19	14	ad2antrr 730	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ a = A ) /\ b = B ) -> C e. _V )
20		simpr 462	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( a = A /\ b = B /\ c = C ) )
21	20, 6	sylibr 215	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> <. a , b , c >. = <. A , B , C >. )
22	21	eqcomd 2437	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> <. A , B , C >. = <. a , b , c >. )
23	1	biimpar 487	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ps )
24	22, 23	jca 534	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
25		a1tru 1453	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> T. )
26	24, 25	2thd 243	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( a = A /\ b = B /\ c = C ) ) -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
27	26	3anassrs 1228	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ a = A ) /\ b = B ) /\ c = C ) -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
28	19, 27	sbcied 3342	. . . . . . . . . . . . 13 |- ( ( ( ph /\ a = A ) /\ b = B ) -> ( [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
29	18, 28	sbcied 3342	. . . . . . . . . . . 12 |- ( ( ph /\ a = A ) -> ( [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
30	17, 29	sbcied 3342	. . . . . . . . . . 11 |- ( ph -> ( [. A / a ]. [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> T. ) )
31	16, 30	mpbiri 236	. . . . . . . . . 10 |- ( ph -> [. A / a ]. [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
32	31	spesbcd 3388	. . . . . . . . 9 |- ( ph -> E. a [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
33		nfcv 2591	. . . . . . . . . 10 |- F/_ b B
34		nfsbc1v 3325	. . . . . . . . . . 11 |- F/ b [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
35	34	nfex 2006	. . . . . . . . . 10 |- F/ b E. a [. B / b ]. [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
36		sbceq1a 3316	. . . . . . . . . . 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 1761	. . . . . . . . . 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 3168	. . . . . . . . 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 62	. . . . . . . 8 |- ( ph -> E. b E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
40		nfcv 2591	. . . . . . . . 9 |- F/_ c C
41		nfsbc1v 3325	. . . . . . . . . . 11 |- F/ c [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
42	41	nfex 2006	. . . . . . . . . 10 |- F/ c E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
43	42	nfex 2006	. . . . . . . . 9 |- F/ c E. b E. a [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps )
44		sbceq1a 3316	. . . . . . . . . 10 |- ( c = C -> ( ( <. A , B , C >. = <. a , b , c >. /\ ps ) <-> [. C / c ]. ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
45	44	2exbidv 1763	. . . . . . . . 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 3168	. . . . . . . 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 62	. . . . . . 7 |- ( ph -> E. c E. b E. a ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
48		excom13 1904	. . . . . . 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 199	. . . . . 6 |- ( ph -> E. a E. b E. c ( <. A , B , C >. = <. a , b , c >. /\ ps ) )
50		eqeq1 2433	. . . . . . . 8 |- ( x = <. A , B , C >. -> ( x = <. a , b , c >. <-> <. A , B , C >. = <. a , b , c >. ) )
51	50	anbi1d 709	. . . . . . 7 |- ( x = <. A , B , C >. -> ( ( x = <. a , b , c >. /\ ps ) <-> ( <. A , B , C >. = <. a , b , c >. /\ ps ) ) )
52	51	3exbidv 1764	. . . . . 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 225	. . . . 5 |- ( ph -> ( x = <. A , B , C >. -> E. a E. b E. c ( x = <. a , b , c >. /\ ps ) ) )
54	13, 53	impbid 193	. . . 4 |- ( ph -> ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = <. A , B , C >. ) )
55	54	alrimiv 1766	. . 3 |- ( ph -> A. x ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = <. A , B , C >. ) )
56		otex 4687	. . . 4 |- <. A , B , C >. e. _V
57		eqeq2 2444	. . . . . 6 |- ( y = <. A , B , C >. -> ( x = y <-> x = <. A , B , C >. ) )
58	57	bibi2d 319	. . . . 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 1760	. . . 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 3179	. . 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 17	. 2 |- ( ph -> E. y A. x ( E. a E. b E. c ( x = <. a , b , c >. /\ ps ) <-> x = y ) )
62		df-eu 2270	. 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 215	1 |- ( ph -> E! x E. a E. b E. c ( x = <. a , b , c >. /\ ps ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   A.wal 1435   = wceq 1437   T. wtru 1438   E.wex 1659   e. wcel 1870   E!weu 2266   _Vcvv 3087   [.wsbc 3305   <.cotp 4010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-ot 4011

This theorem is referenced by:  oeeu  7312