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Theorem otth 4225
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 3591 . . 3  |-  <. A ,  B ,  R >.  = 
<. <. A ,  B >. ,  R >.
2 df-ot 3591 . . 3  |-  <. C ,  D ,  S >.  = 
<. <. C ,  D >. ,  S >.
31, 2eqeq12i 2184 . 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 4224 . 2  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S ) )
83, 7bitri 183 1  |-  ( <. A ,  B ,  R >.  =  <. C ,  D ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 104    /\ w3a 973    = wceq 1348    e. wcel 2141   _Vcvv 2730   <.cop 3584   <.cotp 3585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-ot 3591
This theorem is referenced by:  euotd  4237
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