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Theorem oteq2 4883
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 4874 . . 3 (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩)
21opeq1d 4879 . 2 (𝐴 = 𝐵 → ⟨⟨𝐶, 𝐴⟩, 𝐷⟩ = ⟨⟨𝐶, 𝐵⟩, 𝐷⟩)
3 df-ot 4635 . 2 𝐶, 𝐴, 𝐷⟩ = ⟨⟨𝐶, 𝐴⟩, 𝐷
4 df-ot 4635 . 2 𝐶, 𝐵, 𝐷⟩ = ⟨⟨𝐶, 𝐵⟩, 𝐷
52, 3, 43eqtr4g 2802 1 (𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cop 4632  cotp 4634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635
This theorem is referenced by:  oteq2d  4886  frxp3  8176  xpord3pred  8177  efgi  19737  efgtf  19740  efgtval  19741
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