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Theorem oteq123d 4799
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 4796 . 2 (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐶, 𝐸⟩)
3 oteq123d.2 . . 3 (𝜑𝐶 = 𝐷)
43oteq2d 4797 . 2 (𝜑 → ⟨𝐵, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐸⟩)
5 oteq123d.3 . . 3 (𝜑𝐸 = 𝐹)
65oteq3d 4798 . 2 (𝜑 → ⟨𝐵, 𝐷, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
72, 4, 63eqtrd 2863 1 (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  cotp 4556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-ot 4557
This theorem is referenced by:  idaval  17307  coaval  17317  matval  21006  msrval  32803  mclsax  32834  elmpps  32838  mthmpps  32847  ackval0012  44933  ackval1012  44934  ackval2012  44935  ackval3012  44936
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