Theorem opeq12

Description: Equality theorem for ordered pairs. (Contributed by NM, 28-May-1995)

	Ref	Expression
Assertion	opeq12	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ )

Proof

Step	Hyp	Ref	Expression
1		opeq1	⊢ ( 𝐴 = 𝐶 → ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐶 , 𝐵 ⟩ )
2		opeq2	⊢ ( 𝐵 = 𝐷 → ⟨ 𝐶 , 𝐵 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ )
3	1 2	sylan9eq	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ )