ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  oteq3 Unicode version

Theorem oteq3 3686
Description: Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
Assertion
Ref Expression
oteq3  |-  ( A  =  B  ->  <. C ,  D ,  A >.  = 
<. C ,  D ,  B >. )

Proof of Theorem oteq3
StepHypRef Expression
1 opeq2 3676 . 2  |-  ( A  =  B  ->  <. <. C ,  D >. ,  A >.  = 
<. <. C ,  D >. ,  B >. )
2 df-ot 3507 . 2  |-  <. C ,  D ,  A >.  = 
<. <. C ,  D >. ,  A >.
3 df-ot 3507 . 2  |-  <. C ,  D ,  B >.  = 
<. <. C ,  D >. ,  B >.
41, 2, 33eqtr4g 2175 1  |-  ( A  =  B  ->  <. C ,  D ,  A >.  = 
<. C ,  D ,  B >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316   <.cop 3500   <.cotp 3501
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-ot 3507
This theorem is referenced by:  oteq3d  3689
  Copyright terms: Public domain W3C validator