Theorem ineq12d

Description: Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Hypotheses	ineq1d.1	$$⊢\phi \to A=B$$
	ineq12d.2	$$⊢\phi \to C=D$$
Assertion	ineq12d	$$⊢\phi \to A\cap C=B\cap D$$

Proof

Step	Hyp	Ref	Expression
1		ineq1d.1	$$⊢\phi \to A=B$$
2		ineq12d.2	$$⊢\phi \to C=D$$
3		ineq12	$$⊢\left(A=B\wedge C=D\right)\to A\cap C=B\cap D$$
4	1 2 3	syl2anc	$$⊢\phi \to A\cap C=B\cap D$$