Theorem ineq2

Description: Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993)

	Ref	Expression
Assertion	ineq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ∩ 𝐴 ) = ( 𝐶 ∩ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ineq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∩ 𝐶 ) = ( 𝐵 ∩ 𝐶 ) )
2		incom	⊢ ( 𝐶 ∩ 𝐴 ) = ( 𝐴 ∩ 𝐶 )
3		incom	⊢ ( 𝐶 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐶 )
4	1 2 3	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ∩ 𝐴 ) = ( 𝐶 ∩ 𝐵 ) )