Theorem ineq2d

Description: Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994)

	Ref	Expression
Hypothesis	ineq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	ineq2d	⊢ ( 𝜑 → ( 𝐶 ∩ 𝐴 ) = ( 𝐶 ∩ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ineq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		ineq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ∩ 𝐴 ) = ( 𝐶 ∩ 𝐵 ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐶 ∩ 𝐴 ) = ( 𝐶 ∩ 𝐵 ) )