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Theorem oteq123d 4810
Description: Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
oteq1d.1 (𝜑𝐴 = 𝐵)
oteq123d.2 (𝜑𝐶 = 𝐷)
oteq123d.3 (𝜑𝐸 = 𝐹)
Assertion
Ref Expression
oteq123d (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)

Proof of Theorem oteq123d
StepHypRef Expression
1 oteq1d.1 . . 3 (𝜑𝐴 = 𝐵)
21oteq1d 4807 . 2 (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐶, 𝐸⟩)
3 oteq123d.2 . . 3 (𝜑𝐶 = 𝐷)
43oteq2d 4808 . 2 (𝜑 → ⟨𝐵, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐸⟩)
5 oteq123d.3 . . 3 (𝜑𝐸 = 𝐹)
65oteq3d 4809 . 2 (𝜑 → ⟨𝐵, 𝐷, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
72, 4, 63eqtrd 2857 1 (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  cotp 4565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-ot 4566
This theorem is referenced by:  idaval  17306  coaval  17316  matval  20948  msrval  32682  mclsax  32713  elmpps  32717  mthmpps  32726
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