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Theorem eqvinopb 25068
Description: A variable introduction law for ordered triples. See eqvinop 4267. (Contributed by FL, 6-Nov-2013.)
Hypotheses
Ref Expression
eqvinopb.1  |-  B  e. 
_V
eqvinopb.2  |-  C  e. 
_V
eqvinopb.3  |-  D  e. 
_V
Assertion
Ref Expression
eqvinopb  |-  ( A  =  <. <. B ,  C >. ,  D >.  <->  E. x E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  <. <. x ,  y
>. ,  z >.  = 
<. <. B ,  C >. ,  D >. )
)
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    x, D, y, z

Proof of Theorem eqvinopb
StepHypRef Expression
1 vex 2804 . . . . . . . . . 10  |-  x  e. 
_V
2 vex 2804 . . . . . . . . . 10  |-  y  e. 
_V
3 vex 2804 . . . . . . . . . 10  |-  z  e. 
_V
41, 2, 3otth2 4265 . . . . . . . . 9  |-  ( <. <. x ,  y >. ,  z >.  =  <. <. B ,  C >. ,  D >.  <->  ( x  =  B  /\  y  =  C  /\  z  =  D ) )
54anbi2i 675 . . . . . . . 8  |-  ( ( A  =  <. <. x ,  y >. ,  z
>.  /\  <. <. x ,  y
>. ,  z >.  = 
<. <. B ,  C >. ,  D >. )  <->  ( A  =  <. <. x ,  y >. ,  z
>.  /\  ( x  =  B  /\  y  =  C  /\  z  =  D ) ) )
6 ancom 437 . . . . . . . 8  |-  ( ( A  =  <. <. x ,  y >. ,  z
>.  /\  ( x  =  B  /\  y  =  C  /\  z  =  D ) )  <->  ( (
x  =  B  /\  y  =  C  /\  z  =  D )  /\  A  =  <. <.
x ,  y >. ,  z >. )
)
7 df-3an 936 . . . . . . . . . 10  |-  ( ( x  =  B  /\  y  =  C  /\  z  =  D )  <->  ( ( x  =  B  /\  y  =  C )  /\  z  =  D ) )
87anbi1i 676 . . . . . . . . 9  |-  ( ( ( x  =  B  /\  y  =  C  /\  z  =  D )  /\  A  = 
<. <. x ,  y
>. ,  z >. )  <-> 
( ( ( x  =  B  /\  y  =  C )  /\  z  =  D )  /\  A  =  <. <. x ,  y
>. ,  z >. ) )
9 anass 630 . . . . . . . . . 10  |-  ( ( ( x  =  B  /\  y  =  C )  /\  z  =  D )  <->  ( x  =  B  /\  (
y  =  C  /\  z  =  D )
) )
109anbi1i 676 . . . . . . . . 9  |-  ( ( ( ( x  =  B  /\  y  =  C )  /\  z  =  D )  /\  A  =  <. <. x ,  y
>. ,  z >. )  <-> 
( ( x  =  B  /\  ( y  =  C  /\  z  =  D ) )  /\  A  =  <. <. x ,  y >. ,  z
>. ) )
11 anass 630 . . . . . . . . . 10  |-  ( ( ( x  =  B  /\  ( y  =  C  /\  z  =  D ) )  /\  A  =  <. <. x ,  y >. ,  z
>. )  <->  ( x  =  B  /\  ( ( y  =  C  /\  z  =  D )  /\  A  =  <. <.
x ,  y >. ,  z >. )
) )
12 df-3an 936 . . . . . . . . . . . 12  |-  ( ( y  =  C  /\  z  =  D  /\  A  =  <. <. x ,  y >. ,  z
>. )  <->  ( ( y  =  C  /\  z  =  D )  /\  A  =  <. <. x ,  y
>. ,  z >. ) )
1312bicomi 193 . . . . . . . . . . 11  |-  ( ( ( y  =  C  /\  z  =  D )  /\  A  = 
<. <. x ,  y
>. ,  z >. )  <-> 
( y  =  C  /\  z  =  D  /\  A  =  <. <.
x ,  y >. ,  z >. )
)
1413anbi2i 675 . . . . . . . . . 10  |-  ( ( x  =  B  /\  ( ( y  =  C  /\  z  =  D )  /\  A  =  <. <. x ,  y
>. ,  z >. ) )  <->  ( x  =  B  /\  ( y  =  C  /\  z  =  D  /\  A  = 
<. <. x ,  y
>. ,  z >. ) ) )
1511, 14bitri 240 . . . . . . . . 9  |-  ( ( ( x  =  B  /\  ( y  =  C  /\  z  =  D ) )  /\  A  =  <. <. x ,  y >. ,  z
>. )  <->  ( x  =  B  /\  ( y  =  C  /\  z  =  D  /\  A  = 
<. <. x ,  y
>. ,  z >. ) ) )
168, 10, 153bitri 262 . . . . . . . 8  |-  ( ( ( x  =  B  /\  y  =  C  /\  z  =  D )  /\  A  = 
<. <. x ,  y
>. ,  z >. )  <-> 
( x  =  B  /\  ( y  =  C  /\  z  =  D  /\  A  = 
<. <. x ,  y
>. ,  z >. ) ) )
175, 6, 163bitri 262 . . . . . . 7  |-  ( ( A  =  <. <. x ,  y >. ,  z
>.  /\  <. <. x ,  y
>. ,  z >.  = 
<. <. B ,  C >. ,  D >. )  <->  ( x  =  B  /\  ( y  =  C  /\  z  =  D  /\  A  =  <. <.
x ,  y >. ,  z >. )
) )
1817exbii 1572 . . . . . 6  |-  ( E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
<. <. x ,  y
>. ,  z >.  = 
<. <. B ,  C >. ,  D >. )  <->  E. z ( x  =  B  /\  ( y  =  C  /\  z  =  D  /\  A  = 
<. <. x ,  y
>. ,  z >. ) ) )
19 19.42v 1858 . . . . . 6  |-  ( E. z ( x  =  B  /\  ( y  =  C  /\  z  =  D  /\  A  = 
<. <. x ,  y
>. ,  z >. ) )  <->  ( x  =  B  /\  E. z
( y  =  C  /\  z  =  D  /\  A  =  <. <.
x ,  y >. ,  z >. )
) )
20 3anan12 947 . . . . . . . . 9  |-  ( ( y  =  C  /\  z  =  D  /\  A  =  <. <. x ,  y >. ,  z
>. )  <->  ( z  =  D  /\  ( y  =  C  /\  A  =  <. <. x ,  y
>. ,  z >. ) ) )
2120exbii 1572 . . . . . . . 8  |-  ( E. z ( y  =  C  /\  z  =  D  /\  A  = 
<. <. x ,  y
>. ,  z >. )  <->  E. z ( z  =  D  /\  ( y  =  C  /\  A  =  <. <. x ,  y
>. ,  z >. ) ) )
22 eqvinopb.3 . . . . . . . . 9  |-  D  e. 
_V
23 opeq2 3813 . . . . . . . . . . 11  |-  ( z  =  D  ->  <. <. x ,  y >. ,  z
>.  =  <. <. x ,  y >. ,  D >. )
2423eqeq2d 2307 . . . . . . . . . 10  |-  ( z  =  D  ->  ( A  =  <. <. x ,  y >. ,  z
>. 
<->  A  =  <. <. x ,  y >. ,  D >. ) )
2524anbi2d 684 . . . . . . . . 9  |-  ( z  =  D  ->  (
( y  =  C  /\  A  =  <. <.
x ,  y >. ,  z >. )  <->  ( y  =  C  /\  A  =  <. <. x ,  y >. ,  D >. ) ) )
2622, 25ceqsexv 2836 . . . . . . . 8  |-  ( E. z ( z  =  D  /\  ( y  =  C  /\  A  =  <. <. x ,  y
>. ,  z >. ) )  <->  ( y  =  C  /\  A  = 
<. <. x ,  y
>. ,  D >. ) )
2721, 26bitri 240 . . . . . . 7  |-  ( E. z ( y  =  C  /\  z  =  D  /\  A  = 
<. <. x ,  y
>. ,  z >. )  <-> 
( y  =  C  /\  A  =  <. <.
x ,  y >. ,  D >. ) )
2827anbi2i 675 . . . . . 6  |-  ( ( x  =  B  /\  E. z ( y  =  C  /\  z  =  D  /\  A  = 
<. <. x ,  y
>. ,  z >. ) )  <->  ( x  =  B  /\  ( y  =  C  /\  A  =  <. <. x ,  y
>. ,  D >. ) ) )
2918, 19, 283bitri 262 . . . . 5  |-  ( E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
<. <. x ,  y
>. ,  z >.  = 
<. <. B ,  C >. ,  D >. )  <->  ( x  =  B  /\  ( y  =  C  /\  A  =  <. <.
x ,  y >. ,  D >. ) ) )
3029exbii 1572 . . . 4  |-  ( E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  <. <. x ,  y
>. ,  z >.  = 
<. <. B ,  C >. ,  D >. )  <->  E. y ( x  =  B  /\  ( y  =  C  /\  A  =  <. <. x ,  y
>. ,  D >. ) ) )
31 19.42v 1858 . . . 4  |-  ( E. y ( x  =  B  /\  ( y  =  C  /\  A  =  <. <. x ,  y
>. ,  D >. ) )  <->  ( x  =  B  /\  E. y
( y  =  C  /\  A  =  <. <.
x ,  y >. ,  D >. ) ) )
32 eqvinopb.2 . . . . . 6  |-  C  e. 
_V
33 opeq2 3813 . . . . . . . 8  |-  ( y  =  C  ->  <. x ,  y >.  =  <. x ,  C >. )
3433opeq1d 3818 . . . . . . 7  |-  ( y  =  C  ->  <. <. x ,  y >. ,  D >.  =  <. <. x ,  C >. ,  D >. )
3534eqeq2d 2307 . . . . . 6  |-  ( y  =  C  ->  ( A  =  <. <. x ,  y >. ,  D >.  <-> 
A  =  <. <. x ,  C >. ,  D >. ) )
3632, 35ceqsexv 2836 . . . . 5  |-  ( E. y ( y  =  C  /\  A  = 
<. <. x ,  y
>. ,  D >. )  <-> 
A  =  <. <. x ,  C >. ,  D >. )
3736anbi2i 675 . . . 4  |-  ( ( x  =  B  /\  E. y ( y  =  C  /\  A  = 
<. <. x ,  y
>. ,  D >. ) )  <->  ( x  =  B  /\  A  = 
<. <. x ,  C >. ,  D >. )
)
3830, 31, 373bitri 262 . . 3  |-  ( E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  <. <. x ,  y
>. ,  z >.  = 
<. <. B ,  C >. ,  D >. )  <->  ( x  =  B  /\  A  =  <. <. x ,  C >. ,  D >. ) )
3938exbii 1572 . 2  |-  ( E. x E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
<. <. x ,  y
>. ,  z >.  = 
<. <. B ,  C >. ,  D >. )  <->  E. x ( x  =  B  /\  A  = 
<. <. x ,  C >. ,  D >. )
)
40 eqvinopb.1 . . 3  |-  B  e. 
_V
41 opeq1 3812 . . . . 5  |-  ( x  =  B  ->  <. x ,  C >.  =  <. B ,  C >. )
4241opeq1d 3818 . . . 4  |-  ( x  =  B  ->  <. <. x ,  C >. ,  D >.  = 
<. <. B ,  C >. ,  D >. )
4342eqeq2d 2307 . . 3  |-  ( x  =  B  ->  ( A  =  <. <. x ,  C >. ,  D >.  <->  A  =  <. <. B ,  C >. ,  D >. )
)
4440, 43ceqsexv 2836 . 2  |-  ( E. x ( x  =  B  /\  A  = 
<. <. x ,  C >. ,  D >. )  <->  A  =  <. <. B ,  C >. ,  D >. )
4539, 44bitr2i 241 1  |-  ( A  =  <. <. B ,  C >. ,  D >.  <->  E. x E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  <. <. x ,  y
>. ,  z >.  = 
<. <. B ,  C >. ,  D >. )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656
This theorem is referenced by:  copsexgb  25069
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  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
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