Theorem ineq12i

Description: Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004) (Proof shortened by Eric Schmidt, 26-Jan-2007)

	Ref	Expression
Hypotheses	ineq1i.1	$$⊢A=B$$
	ineq12i.2	$$⊢C=D$$
Assertion	ineq12i	$$⊢A\cap C=B\cap D$$

Proof

Step	Hyp	Ref	Expression
1		ineq1i.1	$$⊢A=B$$
2		ineq12i.2	$$⊢C=D$$
3		ineq12	$$⊢\left(A=B\wedge C=D\right)\to A\cap C=B\cap D$$
4	1 2 3	mp2an	$$⊢A\cap C=B\cap D$$