Theorem eqeq12

Description: Equality relationship among four classes. (Contributed by NM, 3-Aug-1994) (Proof shortened by Wolf Lammen, 23-Oct-2024)

	Ref	Expression
Assertion	eqeq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 )
2		id	⊢ ( 𝐶 = 𝐷 → 𝐶 = 𝐷 )
3	1 2	eqeqan12d	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )