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Theorem otth2 4474
Description: Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (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
otth2  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S ) )

Proof of Theorem otth2
StepHypRef Expression
1 otth.1 . . . 4  |-  A  e. 
_V
2 otth.2 . . . 4  |-  B  e. 
_V
31, 2opth 4470 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  ( A  =  C  /\  B  =  D )
)
43anbi1i 678 . 2  |-  ( (
<. A ,  B >.  = 
<. C ,  D >.  /\  R  =  S )  <-> 
( ( A  =  C  /\  B  =  D )  /\  R  =  S ) )
5 opex 4462 . . 3  |-  <. A ,  B >.  e.  _V
6 otth.3 . . 3  |-  R  e. 
_V
75, 6opth 4470 . 2  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( <. A ,  B >.  =  <. C ,  D >.  /\  R  =  S ) )
8 df-3an 939 . 2  |-  ( ( A  =  C  /\  B  =  D  /\  R  =  S )  <->  ( ( A  =  C  /\  B  =  D )  /\  R  =  S ) )
94, 7, 83bitr4i 270 1  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728   _Vcvv 2965   <.cop 3846
This theorem is referenced by:  otth  4475  oprabid  6141  eloprabga  6196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852
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