MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  otth2 Unicode version

Theorem otth2 4249
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 4245 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  ( A  =  C  /\  B  =  D )
)
43anbi1i 676 . 2  |-  ( (
<. A ,  B >.  = 
<. C ,  D >.  /\  R  =  S )  <-> 
( ( A  =  C  /\  B  =  D )  /\  R  =  S ) )
5 opex 4237 . . 3  |-  <. A ,  B >.  e.  _V
6 otth.3 . . 3  |-  R  e. 
_V
75, 6opth 4245 . 2  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( <. A ,  B >.  =  <. C ,  D >.  /\  R  =  S ) )
8 df-3an 936 . 2  |-  ( ( A  =  C  /\  B  =  D  /\  R  =  S )  <->  ( ( A  =  C  /\  B  =  D )  /\  R  =  S ) )
94, 7, 83bitr4i 268 1  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643
This theorem is referenced by:  otth  4250  oprabid  5882  eloprabga  5934  eqvinopb  24965  copsexgb  24966
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
  Copyright terms: Public domain W3C validator