Theorem oteq123d 4912

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 4909	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐶, 𝐸⟩)
3		oteq123d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	oteq2d 4910	. 2 ⊢ (𝜑 → ⟨𝐵, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐸⟩)
5		oteq123d.3	. . 3 ⊢ (𝜑 → 𝐸 = 𝐹)
6	5	oteq3d 4911	. 2 ⊢ (𝜑 → ⟨𝐵, 𝐷, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)
7	2, 4, 6	3eqtrd 2784	1 ⊢ (𝜑 → ⟨𝐴, 𝐶, 𝐸⟩ = ⟨𝐵, 𝐷, 𝐹⟩)

Syntax hints:   → wi 4   = wceq 1537  ⟨cotp 4656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-ot 4657

This theorem is referenced by:  idaval  18125  coaval  18135  matval  22436  msrval  35506  mclsax  35537  elmpps  35541  mthmpps  35550  ackval0012  48423  ackval1012  48424  ackval2012  48425  ackval3012  48426  mndtcval  48752