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Theorem oteq2 3710
 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 3701 . . 3 (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
21opeq1d 3706 . 2 (𝐴 = 𝐵 → ⟨⟨𝐶, 𝐴⟩, 𝐷⟩ = ⟨⟨𝐶, 𝐵⟩, 𝐷⟩)
3 df-ot 3532 . 2 𝐶, 𝐴, 𝐷⟩ = ⟨⟨𝐶, 𝐴⟩, 𝐷
4 df-ot 3532 . 2 𝐶, 𝐵, 𝐷⟩ = ⟨⟨𝐶, 𝐵⟩, 𝐷
52, 3, 43eqtr4g 2195 1 (𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331  ⟨cop 3525  ⟨cotp 3526 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-ot 3532 This theorem is referenced by:  oteq2d  3713
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