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Theorem oteq1 4349
 Description: Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
Assertion
Ref Expression
oteq1 (𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)

Proof of Theorem oteq1
StepHypRef Expression
1 opeq1 4340 . . 3 (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
21opeq1d 4346 . 2 (𝐴 = 𝐵 → ⟨⟨𝐴, 𝐶⟩, 𝐷⟩ = ⟨⟨𝐵, 𝐶⟩, 𝐷⟩)
3 df-ot 4134 . 2 𝐴, 𝐶, 𝐷⟩ = ⟨⟨𝐴, 𝐶⟩, 𝐷
4 df-ot 4134 . 2 𝐵, 𝐶, 𝐷⟩ = ⟨⟨𝐵, 𝐶⟩, 𝐷
52, 3, 43eqtr4g 2669 1 (𝐴 = 𝐵 → ⟨𝐴, 𝐶, 𝐷⟩ = ⟨𝐵, 𝐶, 𝐷⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ⟨cop 4131  ⟨cotp 4133 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134 This theorem is referenced by:  oteq1d  4352  otiunsndisj  4905  efgi  17955  efgtf  17958  efgtval  17959  mapdh9a  36097  mapdh9aOLDN  36098  hdmapval2  36142  otiunsndisjX  40317
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