Theorem oteq123d 4841

Description: Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
oteq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
oteq123d.2	⊢ (𝜑 → 𝐶 = 𝐷)
oteq123d.3	⊢ (𝜑 → 𝐸 = 𝐹)

Assertion

Ref	Expression
oteq123d	⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)

Proof of Theorem oteq123d

Step	Hyp	Ref	Expression
1		oteq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	oteq1d 4838	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐶, 𝐸⟩)
3		oteq123d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	oteq2d 4839	. 2 ⊢ (𝜑 → ⟨𝐵, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐸⟩)
5		oteq123d.3	. . 3 ⊢ (𝜑 → 𝐸 = 𝐹)
6	5	oteq3d 4840	. 2 ⊢ (𝜑 → ⟨𝐵, 𝐷, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
7	2, 4, 6	3eqtrd 2772	1 ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)

Syntax hints:   → wi 4   = wceq 1541  ⟨cotp 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-ot 4586

This theorem is referenced by:  idaval  17973  coaval  17983  matval  22346  msrval  35654  mclsax  35685  elmpps  35689  mthmpps  35698  ackval0012  48851  ackval1012  48852  ackval2012  48853  ackval3012  48854  termcarweu  49689  mndtcval  49740