Theorem ineqan12d

Description: Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007)

	Ref	Expression
Hypotheses	ineq1d.1	$$⊢\phi \to A=B$$
	ineqan12d.2	$$⊢\psi \to C=D$$
Assertion	ineqan12d	$$⊢\left(\phi \wedge \psi \right)\to A\cap C=B\cap D$$

Proof

Step	Hyp	Ref	Expression
1		ineq1d.1	$$⊢\phi \to A=B$$
2		ineqan12d.2	$$⊢\psi \to C=D$$
3		ineq12	$$⊢\left(A=B\wedge C=D\right)\to A\cap C=B\cap D$$
4	1 2 3	syl2an	$$⊢\left(\phi \wedge \psi \right)\to A\cap C=B\cap D$$