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Theorem otth 4250
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 3650 . . 3  |-  <. A ,  B ,  R >.  = 
<. <. A ,  B >. ,  R >.
2 df-ot 3650 . . 3  |-  <. C ,  D ,  S >.  = 
<. <. C ,  D >. ,  S >.
31, 2eqeq12i 2296 . 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 4249 . 2  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S ) )
83, 7bitri 240 1  |-  ( <. A ,  B ,  R >.  =  <. C ,  D ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   <.cotp 3644
This theorem is referenced by:  euotd  4267
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650
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