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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.
StepHypRef Expression
1 euotd.4 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ps  <->  ( a  =  A  /\  b  =  B  /\  c  =  C ) ) )
21biimpa 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
63, 4, 5otth 4722 . . . . . . . . . . . 12  |-  ( <.
a ,  b ,  c >.  =  <. A ,  B ,  C >.  <-> 
( a  =  A  /\  b  =  B  /\  c  =  C ) )
72, 6sylibr 212 . . . . . . . . . . 11  |-  ( (
ph  /\  ps )  -> 
<. a ,  b ,  c >.  =  <. A ,  B ,  C >. )
87eqeq2d 2474 . . . . . . . . . 10  |-  ( (
ph  /\  ps )  ->  ( x  =  <. a ,  b ,  c
>. 
<->  x  =  <. A ,  B ,  C >. ) )
98biimpd 207 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  ( x  =  <. a ,  b ,  c
>.  ->  x  =  <. A ,  B ,  C >. ) )
109impancom 440 . . . . . . . 8  |-  ( (
ph  /\  x  =  <. a ,  b ,  c >. )  ->  ( ps  ->  x  =  <. A ,  B ,  C >. ) )
1110expimpd 603 . . . . . . 7  |-  ( ph  ->  ( ( x  = 
<. a ,  b ,  c >.  /\  ps )  ->  x  =  <. A ,  B ,  C >. ) )
1211exlimdv 1695 . . . . . 6  |-  ( ph  ->  ( E. c ( x  =  <. a ,  b ,  c
>.  /\  ps )  ->  x  =  <. A ,  B ,  C >. ) )
1312exlimdvv 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 )
1815adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  =  A )  ->  B  e.  _V )
1914ad2antrr 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 ) )
2120, 6sylibr 212 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  ->  <. a ,  b ,  c >.  =  <. A ,  B ,  C >. )
2221eqcomd 2468 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  ->  <. A ,  B ,  C >.  =  <. a ,  b ,  c
>. )
231biimpar 485 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  ->  ps )
2422, 23jca 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.  )
2624, 252thd 240 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  -> 
( ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )  <-> T.  ) )
27263anassrs 1213 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  =  A )  /\  b  =  B
)  /\  c  =  C )  ->  (
( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <-> T.  )
)
2819, 27sbcied 3361 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  =  A )  /\  b  =  B )  ->  ( [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <-> T.  )
)
2918, 28sbcied 3361 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  =  A )  ->  ( [. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <-> T.  )
)
3017, 29sbcied 3361 . . . . . . . . . . 11  |-  ( ph  ->  ( [. A  / 
a ]. [. B  / 
b ]. [. C  / 
c ]. ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )  <-> T.  ) )
3116, 30mpbiri 233 . . . . . . . . . 10  |-  ( ph  ->  [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps ) )
3231spesbcd 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 )
3534nfex 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 )
) )
3736exbidv 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 )
) )
3833, 35, 37spcegf 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 ) ) )
3915, 32, 38sylc 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 )
4241nfex 1890 . . . . . . . . . 10  |-  F/ c E. a [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )
4342nfex 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 )
) )
45442exbidv 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 )
) )
4640, 43, 45spcegf 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 )
) )
4714, 39, 46sylc 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 )
)
4947, 48sylib 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 >. ) )
5150anbi1d 704 . . . . . . 7  |-  ( x  =  <. A ,  B ,  C >.  ->  ( ( x  =  <. a ,  b ,  c
>.  /\  ps )  <->  ( <. A ,  B ,  C >.  =  <. a ,  b ,  c >.  /\  ps ) ) )
52513exbidv 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 )
) )
5349, 52syl5ibrcom 222 . . . . 5  |-  ( ph  ->  ( x  =  <. A ,  B ,  C >.  ->  E. a E. b E. c ( x  = 
<. a ,  b ,  c >.  /\  ps )
) )
5413, 53impbid 191 . . . 4  |-  ( ph  ->  ( E. a E. b E. c ( x  =  <. a ,  b ,  c
>.  /\  ps )  <->  x  =  <. A ,  B ,  C >. ) )
5554alrimiv 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 >. ) )
5857bibi2d 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 >. ) ) )
5958albidv 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 >. ) ) )
6056, 59spcev 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 ) )
6155, 60syl 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 ) )
6361, 62sylibr 212 1  |-  ( ph  ->  E! x E. a E. b E. c ( x  =  <. a ,  b ,  c
>.  /\  ps ) )
Colors of variables: wff setvar class
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
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