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Theorem oteq123d 3808
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 3805 . 2 (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐶, 𝐸⟩)
3 oteq123d.2 . . 3 (𝜑𝐶 = 𝐷)
43oteq2d 3806 . 2 (𝜑 → ⟨𝐵, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐸⟩)
5 oteq123d.3 . . 3 (𝜑𝐸 = 𝐹)
65oteq3d 3807 . 2 (𝜑 → ⟨𝐵, 𝐷, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
72, 4, 63eqtrd 2226 1 (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  cotp 3611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-ot 3617
This theorem is referenced by: (None)
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