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Theorem oteq3 4604
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 4594 . 2 (𝐴 = 𝐵 → ⟨⟨𝐶, 𝐷⟩, 𝐴⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐵⟩)
2 df-ot 4377 . 2 𝐶, 𝐷, 𝐴⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐴
3 df-ot 4377 . 2 𝐶, 𝐷, 𝐵⟩ = ⟨⟨𝐶, 𝐷⟩, 𝐵
41, 2, 33eqtr4g 2858 1 (𝐴 = 𝐵 → ⟨𝐶, 𝐷, 𝐴⟩ = ⟨𝐶, 𝐷, 𝐵⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  cop 4374  cotp 4376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-ot 4377
This theorem is referenced by:  oteq3d  4607  otsndisj  5175  otiunsndisj  5176  efgi0  18446  efgi1  18447  mapdhcl  37748  mapdh6dN  37760  mapdh8  37809  mapdh9a  37810  mapdh9aOLDN  37811  hdmap1l6d  37834  hdmapval  37849  hdmapval2  37853  hdmapval3N  37859  otiunsndisjX  42134
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