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Theorem oteq123d 4568
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 4565 . 2 (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐶, 𝐸⟩)
3 oteq123d.2 . . 3 (𝜑𝐶 = 𝐷)
43oteq2d 4566 . 2 (𝜑 → ⟨𝐵, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐸⟩)
5 oteq123d.3 . . 3 (𝜑𝐸 = 𝐹)
65oteq3d 4567 . 2 (𝜑 → ⟨𝐵, 𝐷, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
72, 4, 63eqtrd 2798 1 (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  cotp 4329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-ot 4330
This theorem is referenced by:  idaval  16909  coaval  16919  matval  20419  msrval  31742  mclsax  31773  elmpps  31777  mthmpps  31786
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