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Theorem otth 4432
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 3816 . . 3  |-  <. A ,  B ,  R >.  = 
<. <. A ,  B >. ,  R >.
2 df-ot 3816 . . 3  |-  <. C ,  D ,  S >.  = 
<. <. C ,  D >. ,  S >.
31, 2eqeq12i 2448 . 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 4431 . 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 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   <.cotp 3810
This theorem is referenced by:  euotd  4449  2spotiundisj  28388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-ot 3816
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