Theorem eqeq12

Description: Equality relationship among four classes. (Contributed by NM, 3-Aug-1994)

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

Proof

Step	Hyp	Ref	Expression
1		eqeq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐶 ) )
2		eqeq2	⊢ ( 𝐶 = 𝐷 → ( 𝐵 = 𝐶 ↔ 𝐵 = 𝐷 ) )
3	1 2	sylan9bb	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐷 ) )