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Theorem euotd 4283
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 470 . . . . . . . . . . . 12  |-  ( (
ph  /\  ps )  ->  ( a  =  A  /\  b  =  B  /\  c  =  C ) )
3 vex 2804 . . . . . . . . . . . . 13  |-  a  e. 
_V
4 vex 2804 . . . . . . . . . . . . 13  |-  b  e. 
_V
5 vex 2804 . . . . . . . . . . . . 13  |-  c  e. 
_V
63, 4, 5otth 4266 . . . . . . . . . . . 12  |-  ( <.
a ,  b ,  c >.  =  <. A ,  B ,  C >.  <-> 
( a  =  A  /\  b  =  B  /\  c  =  C ) )
72, 6sylibr 203 . . . . . . . . . . 11  |-  ( (
ph  /\  ps )  -> 
<. a ,  b ,  c >.  =  <. A ,  B ,  C >. )
87eqeq2d 2307 . . . . . . . . . 10  |-  ( (
ph  /\  ps )  ->  ( x  =  <. a ,  b ,  c
>. 
<->  x  =  <. A ,  B ,  C >. ) )
98biimpd 198 . . . . . . . . 9  |-  ( (
ph  /\  ps )  ->  ( x  =  <. a ,  b ,  c
>.  ->  x  =  <. A ,  B ,  C >. ) )
109impancom 427 . . . . . . . 8  |-  ( (
ph  /\  x  =  <. a ,  b ,  c >. )  ->  ( ps  ->  x  =  <. A ,  B ,  C >. ) )
1110expimpd 586 . . . . . . 7  |-  ( ph  ->  ( ( x  = 
<. a ,  b ,  c >.  /\  ps )  ->  x  =  <. A ,  B ,  C >. ) )
1211exlimdv 1626 . . . . . 6  |-  ( ph  ->  ( E. c ( x  =  <. a ,  b ,  c
>.  /\  ps )  ->  x  =  <. A ,  B ,  C >. ) )
1312exlimdvv 1627 . . . . 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 euotd.1 . . . . . . . . . 10  |-  ( ph  ->  A  e.  _V )
17 tru 1312 . . . . . . . . . . 11  |-  T.
1815adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  a  =  A )  ->  B  e.  _V )
1914ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  a  =  A )  /\  b  =  B )  ->  C  e.  _V )
20 simpr 447 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  -> 
( a  =  A  /\  b  =  B  /\  c  =  C ) )
2120, 6sylibr 203 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  ->  <. a ,  b ,  c >.  =  <. A ,  B ,  C >. )
2221eqcomd 2301 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  ->  <. A ,  B ,  C >.  =  <. a ,  b ,  c
>. )
231biimpar 471 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  ->  ps )
2422, 23jca 518 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  -> 
( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps ) )
25 a1tru 1321 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  ->  T.  )
2624, 252thd 231 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( a  =  A  /\  b  =  B  /\  c  =  C ) )  -> 
( ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )  <->  T.  ) )
27263exp2 1169 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( a  =  A  ->  ( b  =  B  ->  ( c  =  C  ->  ( (
<. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <->  T.  )
) ) ) )
2827imp41 576 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  =  A )  /\  b  =  B
)  /\  c  =  C )  ->  (
( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <->  T.  )
)
2919, 28sbcied 3040 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  =  A )  /\  b  =  B )  ->  ( [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <->  T.  )
)
3018, 29sbcied 3040 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  =  A )  ->  ( [. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <->  T.  )
)
3116, 30sbcied 3040 . . . . . . . . . . 11  |-  ( ph  ->  ( [. A  / 
a ]. [. B  / 
b ]. [. C  / 
c ]. ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )  <->  T.  ) )
3217, 31mpbiri 224 . . . . . . . . . 10  |-  ( ph  ->  [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps ) )
33 nfcv 2432 . . . . . . . . . . 11  |-  F/_ a A
34 nfsbc1v 3023 . . . . . . . . . . 11  |-  F/ a
[. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )
35 sbceq1a 3014 . . . . . . . . . . 11  |-  ( a  =  A  ->  ( [. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <->  [. A  / 
a ]. [. B  / 
b ]. [. C  / 
c ]. ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )
) )
3633, 34, 35spcegf 2877 . . . . . . . . . 10  |-  ( A  e.  _V  ->  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  ->  E. a [. B  / 
b ]. [. C  / 
c ]. ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )
) )
3716, 32, 36sylc 56 . . . . . . . . 9  |-  ( ph  ->  E. a [. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps ) )
38 nfcv 2432 . . . . . . . . . 10  |-  F/_ b B
39 nfsbc1v 3023 . . . . . . . . . . 11  |-  F/ b
[. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )
4039nfex 1779 . . . . . . . . . 10  |-  F/ b E. a [. B  /  b ]. [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )
41 sbceq1a 3014 . . . . . . . . . . 11  |-  ( b  =  B  ->  ( [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <->  [. B  / 
b ]. [. C  / 
c ]. ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )
) )
4241exbidv 1616 . . . . . . . . . 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 )
) )
4338, 40, 42spcegf 2877 . . . . . . . . 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 ) ) )
4415, 37, 43sylc 56 . . . . . . . 8  |-  ( ph  ->  E. b E. a [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps ) )
45 nfcv 2432 . . . . . . . . 9  |-  F/_ c C
46 nfsbc1v 3023 . . . . . . . . . . 11  |-  F/ c
[. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )
4746nfex 1779 . . . . . . . . . 10  |-  F/ c E. a [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )
4847nfex 1779 . . . . . . . . 9  |-  F/ c E. b E. a [. C  /  c ]. ( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )
49 sbceq1a 3014 . . . . . . . . . 10  |-  ( c  =  C  ->  (
( <. A ,  B ,  C >.  =  <. a ,  b ,  c
>.  /\  ps )  <->  [. C  / 
c ]. ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )
) )
50492exbidv 1618 . . . . . . . . 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 )
) )
5145, 48, 50spcegf 2877 . . . . . . . 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 )
) )
5214, 44, 51sylc 56 . . . . . . 7  |-  ( ph  ->  E. c E. b E. a ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )
)
53 excom13 1829 . . . . . . 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 )
)
5452, 53sylib 188 . . . . . 6  |-  ( ph  ->  E. a E. b E. c ( <. A ,  B ,  C >.  = 
<. a ,  b ,  c >.  /\  ps )
)
55 eqeq1 2302 . . . . . . . 8  |-  ( x  =  <. A ,  B ,  C >.  ->  ( x  =  <. a ,  b ,  c >.  <->  <. A ,  B ,  C >.  = 
<. a ,  b ,  c >. ) )
5655anbi1d 685 . . . . . . 7  |-  ( x  =  <. A ,  B ,  C >.  ->  ( ( x  =  <. a ,  b ,  c
>.  /\  ps )  <->  ( <. A ,  B ,  C >.  =  <. a ,  b ,  c >.  /\  ps ) ) )
57563exbidv 1619 . . . . . 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 )
) )
5854, 57syl5ibrcom 213 . . . . 5  |-  ( ph  ->  ( x  =  <. A ,  B ,  C >.  ->  E. a E. b E. c ( x  = 
<. a ,  b ,  c >.  /\  ps )
) )
5913, 58impbid 183 . . . 4  |-  ( ph  ->  ( E. a E. b E. c ( x  =  <. a ,  b ,  c
>.  /\  ps )  <->  x  =  <. A ,  B ,  C >. ) )
6059alrimiv 1621 . . 3  |-  ( ph  ->  A. x ( E. a E. b E. c ( x  = 
<. a ,  b ,  c >.  /\  ps )  <->  x  =  <. A ,  B ,  C >. ) )
61 otex 4254 . . . 4  |-  <. A ,  B ,  C >.  e. 
_V
62 eqeq2 2305 . . . . . 6  |-  ( y  =  <. A ,  B ,  C >.  ->  ( x  =  y  <->  x  =  <. A ,  B ,  C >. ) )
6362bibi2d 309 . . . . 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 >. ) ) )
6463albidv 1615 . . . 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 >. ) ) )
6561, 64spcev 2888 . . 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 ) )
6660, 65syl 15 . 2  |-  ( ph  ->  E. y A. x
( E. a E. b E. c ( x  =  <. a ,  b ,  c
>.  /\  ps )  <->  x  =  y ) )
67 df-eu 2160 . 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 ) )
6866, 67sylibr 203 1  |-  ( ph  ->  E! x E. a E. b E. c ( x  =  <. a ,  b ,  c
>.  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    T. wtru 1307   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156   _Vcvv 2801   [.wsbc 3004   <.cotp 3657
This theorem is referenced by:  oeeu  6617
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-ot 3663
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