Theorem ineq12

Description: Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994)

	Ref	Expression
Assertion	ineq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		ineq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) )
2		ineq2	⊢ ( 𝐶 = 𝐷 → ( 𝐵 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐷 ) )
3	1 2	sylan9eq	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐷 ) )