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Theorem oteq2 3559
Description: Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
Assertion
Ref Expression
oteq2  |-  ( A  =  B  ->  <. C ,  A ,  D >.  = 
<. C ,  B ,  D >. )

Proof of Theorem oteq2
StepHypRef Expression
1 opeq2 3550 . . 3  |-  ( A  =  B  ->  <. C ,  A >.  =  <. C ,  B >. )
21opeq1d 3555 . 2  |-  ( A  =  B  ->  <. <. C ,  A >. ,  D >.  = 
<. <. C ,  B >. ,  D >. )
3 df-ot 3385 . 2  |-  <. C ,  A ,  D >.  = 
<. <. C ,  A >. ,  D >.
4 df-ot 3385 . 2  |-  <. C ,  B ,  D >.  = 
<. <. C ,  B >. ,  D >.
52, 3, 43eqtr4g 2097 1  |-  ( A  =  B  ->  <. C ,  A ,  D >.  = 
<. C ,  B ,  D >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243   <.cop 3378   <.cotp 3379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-ot 3385
This theorem is referenced by:  oteq2d  3562
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