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Theorem oteq123d 3587
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 3584 . 2 (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐶, 𝐸⟩)
3 oteq123d.2 . . 3 (𝜑𝐶 = 𝐷)
43oteq2d 3585 . 2 (𝜑 → ⟨𝐵, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐸⟩)
5 oteq123d.3 . . 3 (𝜑𝐸 = 𝐹)
65oteq3d 3586 . 2 (𝜑 → ⟨𝐵, 𝐷, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
72, 4, 63eqtrd 2118 1 (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  cotp 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-ot 3410
This theorem is referenced by: (None)
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