Theorem oteq123d

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

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

Proof

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