Theorem opeq12d

Description: Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006) (Proof shortened by Andrew Salmon, 29-Jun-2011)

	Ref	Expression
Hypotheses	opeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	opeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	opeq12d	⊢ ( 𝜑 → ⟨ 𝐴 , 𝐶 ⟩ = ⟨ 𝐵 , 𝐷 ⟩ )

Proof

Step	Hyp	Ref	Expression
1		opeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		opeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3		opeq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ⟨ 𝐴 , 𝐶 ⟩ = ⟨ 𝐵 , 𝐷 ⟩ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ⟨ 𝐴 , 𝐶 ⟩ = ⟨ 𝐵 , 𝐷 ⟩ )