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Theorem oteq123d 3555
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 3552 . 2 (φ → ⟨A, 𝐶, 𝐸⟩ = ⟨B, 𝐶, 𝐸⟩)
3 oteq123d.2 . . 3 (φ𝐶 = 𝐷)
43oteq2d 3553 . 2 (φ → ⟨B, 𝐶, 𝐸⟩ = ⟨B, 𝐷, 𝐸⟩)
5 oteq123d.3 . . 3 (φ𝐸 = 𝐹)
65oteq3d 3554 . 2 (φ → ⟨B, 𝐷, 𝐸⟩ = ⟨B, 𝐷, 𝐹⟩)
72, 4, 63eqtrd 2073 1 (φ → ⟨A, 𝐶, 𝐸⟩ = ⟨B, 𝐷, 𝐹⟩)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  cotp 3371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-ot 3377
This theorem is referenced by: (None)
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