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Theorem otth 4375
Description: Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
otth.1  |-  A  e. 
_V
otth.2  |-  B  e. 
_V
otth.3  |-  R  e. 
_V
Assertion
Ref Expression
otth  |-  ( <. A ,  B ,  R >.  =  <. C ,  D ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S )
)

Proof of Theorem otth
StepHypRef Expression
1 df-ot 3761 . . 3  |-  <. A ,  B ,  R >.  = 
<. <. A ,  B >. ,  R >.
2 df-ot 3761 . . 3  |-  <. C ,  D ,  S >.  = 
<. <. C ,  D >. ,  S >.
31, 2eqeq12i 2394 . 2  |-  ( <. A ,  B ,  R >.  =  <. C ,  D ,  S >.  <->  <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >. )
4 otth.1 . . 3  |-  A  e. 
_V
5 otth.2 . . 3  |-  B  e. 
_V
6 otth.3 . . 3  |-  R  e. 
_V
74, 5, 6otth2 4374 . 2  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S ) )
83, 7bitri 241 1  |-  ( <. A ,  B ,  R >.  =  <. C ,  D ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2893   <.cop 3754   <.cotp 3755
This theorem is referenced by:  euotd  4392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-nul 4273  ax-pr 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-rab 2652  df-v 2895  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-sn 3757  df-pr 3758  df-op 3760  df-ot 3761
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