ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  otth2 Unicode version

Theorem otth2 4032
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 4028 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  ( A  =  C  /\  B  =  D )
)
43anbi1i 446 . 2  |-  ( (
<. A ,  B >.  = 
<. C ,  D >.  /\  R  =  S )  <-> 
( ( A  =  C  /\  B  =  D )  /\  R  =  S ) )
51, 2opex 4020 . . 3  |-  <. A ,  B >.  e.  _V
6 otth.3 . . 3  |-  R  e. 
_V
75, 6opth 4028 . 2  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( <. A ,  B >.  =  <. C ,  D >.  /\  R  =  S ) )
8 df-3an 922 . 2  |-  ( ( A  =  C  /\  B  =  D  /\  R  =  S )  <->  ( ( A  =  C  /\  B  =  D )  /\  R  =  S ) )
94, 7, 83bitr4i 210 1  |-  ( <. <. A ,  B >. ,  R >.  =  <. <. C ,  D >. ,  S >.  <->  ( A  =  C  /\  B  =  D  /\  R  =  S ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   _Vcvv 2612   <.cop 3425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431
This theorem is referenced by:  otth  4033  oprabid  5616  eloprabga  5670
  Copyright terms: Public domain W3C validator