Theorem opeq12i

Description: Equality inference for ordered pairs. (Contributed by NM, 16-Dec-2006) (Proof shortened by Eric Schmidt, 4-Apr-2007)

	Ref	Expression
Hypotheses	opeq1i.1	⊢ 𝐴 = 𝐵
	opeq12i.2	⊢ 𝐶 = 𝐷
Assertion	opeq12i	⊢ ⟨ 𝐴 , 𝐶 ⟩ = ⟨ 𝐵 , 𝐷 ⟩

Proof

Step	Hyp	Ref	Expression
1		opeq1i.1	⊢ 𝐴 = 𝐵
2		opeq12i.2	⊢ 𝐶 = 𝐷
3		opeq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ⟨ 𝐴 , 𝐶 ⟩ = ⟨ 𝐵 , 𝐷 ⟩ )
4	1 2 3	mp2an	⊢ ⟨ 𝐴 , 𝐶 ⟩ = ⟨ 𝐵 , 𝐷 ⟩