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Theorem euotd 4702
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 492 . . . . . . . . . . . 12  |-  ( (
ph  /\  ps )  ->  ( a  =  A  /\  b  =  B  /\  c  =  C ) )
3 vex 3034 . . . . . . . . . . . . 13  |-  a  e. 
_V
4 vex 3034 . . . . . . . . . . . . 13  |-  b  e. 
_V
5 vex 3034 . . . . . . . . . . . . 13  |-  c  e. 
_V
63, 4, 5otth 4684 . . . . . . . . . . . 12  |-  ( <.
a ,  b ,  c >.  =  <. A ,  B ,  C >.  <-> 
( a  =  A  /\  b  =  B  /\  c  =  C ) )
72, 6sylibr 217 . . . . . . . . . . 11  |-  ( (
ph  /\  ps )  -> 
<. a ,  b ,  c >.  =  <. A ,  B ,  C >. )
87eqeq2d 2481 . . . . . . . . . 10  |-  ( (
ph  /\  ps )  ->  ( x  =  <. a ,  b ,  c
>. 
<->  x  =  <. A ,  B ,  C >. ) )
98biimpd 212 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  ( x  =  <. a ,  b ,  c
>.  ->  x  =  <. A ,  B ,  C >. ) )
109impancom 447 . . . . . . . 8  |-  ( (
ph  /\  x  =  <. a ,  b ,  c >. )  ->  ( ps  ->  x  =  <. A ,  B ,  C >. ) )
1110expimpd 614 . . . . . . 7  |-  ( ph  ->  ( ( x  = 
<. a ,  b ,  c >.  /\  ps )  ->  x  =  <. A ,  B ,  C >. ) )
1211exlimdv 1787 . . . . . 6  |-  ( ph  ->  ( E. c ( x  =  <. a ,  b ,  c
>.  /\  ps )  ->  x  =  <. A ,  B ,  C >. ) )
1312exlimdvv 1788 . . . . 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 1456 . . . . . . . . . . 11  |- T.
17 euotd.1 . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  _V )
1815adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  =  A )  ->  B  e.  _V )
1914ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  =  A )  /\  b  =  B )  ->  C  e.  _V )
20 simpr 468 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  -> 
( a  =  A  /\  b  =  B  /\  c  =  C ) )
2120, 6sylibr 217 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  ->  <. a ,  b ,  c >.  =  <. A ,  B ,  C >. )
2221eqcomd 2477 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  ->  <. A ,  B ,  C >.  =  <. a ,  b ,  c
>. )
231biimpar 493 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  ->  ps )
2422, 23jca 541 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  -> 
( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps ) )
25 a1tru 1468 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  -> T.  )
2624, 252thd 248 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  -> 
( ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )  <-> T.  ) )
27263anassrs 1256 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  =  A )  /\  b  =  B
)  /\  c  =  C )  ->  (
( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <-> T.  )
)
2819, 27sbcied 3292 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  =  A )  /\  b  =  B )  ->  ( [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <-> T.  )
)
2918, 28sbcied 3292 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  =  A )  ->  ( [. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <-> T.  )
)
3017, 29sbcied 3292 . . . . . . . . . . 11  |-  ( ph  ->  ( [. A  / 
a ]. [. B  / 
b ]. [. C  / 
c ]. ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )  <-> T.  ) )
3116, 30mpbiri 241 . . . . . . . . . 10  |-  ( ph  ->  [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps ) )
3231spesbcd 3338 . . . . . . . . 9  |-  ( ph  ->  E. a [. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps ) )
33 nfcv 2612 . . . . . . . . . 10  |-  F/_ b B
34 nfsbc1v 3275 . . . . . . . . . . 11  |-  F/ b
[. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )
3534nfex 2050 . . . . . . . . . 10  |-  F/ b E. a [. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )
36 sbceq1a 3266 . . . . . . . . . . 11  |-  ( b  =  B  ->  ( [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <->  [. B  / 
b ]. [. C  / 
c ]. ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )
) )
3736exbidv 1776 . . . . . . . . . 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 3116 . . . . . . . . 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 61 . . . . . . . 8  |-  ( ph  ->  E. b E. a [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps ) )
40 nfcv 2612 . . . . . . . . 9  |-  F/_ c C
41 nfsbc1v 3275 . . . . . . . . . . 11  |-  F/ c
[. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )
4241nfex 2050 . . . . . . . . . 10  |-  F/ c E. a [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )
4342nfex 2050 . . . . . . . . 9  |-  F/ c E. b E. a [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )
44 sbceq1a 3266 . . . . . . . . . 10  |-  ( c  =  C  ->  (
( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <->  [. C  / 
c ]. ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )
) )
45442exbidv 1778 . . . . . . . . 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 3116 . . . . . . . 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 61 . . . . . . 7  |-  ( ph  ->  E. c E. b E. a ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )
)
48 excom13 1947 . . . . . . 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 201 . . . . . 6  |-  ( ph  ->  E. a E. b E. c ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )
)
50 eqeq1 2475 . . . . . . . 8  |-  ( x  =  <. A ,  B ,  C >.  ->  ( x  =  <. a ,  b ,  c >.  <->  <. A ,  B ,  C >.  = 
<. a ,  b ,  c >. ) )
5150anbi1d 719 . . . . . . 7  |-  ( x  =  <. A ,  B ,  C >.  ->  ( ( x  =  <. a ,  b ,  c
>.  /\  ps )  <->  ( <. A ,  B ,  C >.  =  <. a ,  b ,  c >.  /\  ps ) ) )
52513exbidv 1779 . . . . . 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 230 . . . . 5  |-  ( ph  ->  ( x  =  <. A ,  B ,  C >.  ->  E. a E. b E. c ( x  = 
<. a ,  b ,  c >.  /\  ps )
) )
5413, 53impbid 195 . . . 4  |-  ( ph  ->  ( E. a E. b E. c ( x  =  <. a ,  b ,  c
>.  /\  ps )  <->  x  =  <. A ,  B ,  C >. ) )
5554alrimiv 1781 . . 3  |-  ( ph  ->  A. x ( E. a E. b E. c ( x  = 
<. a ,  b ,  c >.  /\  ps )  <->  x  =  <. A ,  B ,  C >. ) )
56 otex 4665 . . . 4  |-  <. A ,  B ,  C >.  e. 
_V
57 eqeq2 2482 . . . . . 6  |-  ( y  =  <. A ,  B ,  C >.  ->  ( x  =  y  <->  x  =  <. A ,  B ,  C >. ) )
5857bibi2d 325 . . . . 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 1775 . . . 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 3127 . . 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 17 . 2  |-  ( ph  ->  E. y A. x
( E. a E. b E. c ( x  =  <. a ,  b ,  c
>.  /\  ps )  <->  x  =  y ) )
62 df-eu 2323 . 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 217 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 189    /\ wa 376    /\ w3a 1007   A.wal 1450    = wceq 1452   T. wtru 1453   E.wex 1671    e. wcel 1904   E!weu 2319   _Vcvv 3031   [.wsbc 3255   <.cotp 3967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-ot 3968
This theorem is referenced by:  oeeu  7322
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