Theorem eleqtrd

Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004)

	Ref	Expression
Hypotheses	eleqtrd.1	$$⊢\phi \to A\in B$$
	eleqtrd.2	$$⊢\phi \to B=C$$
Assertion	eleqtrd	$$⊢\phi \to A\in C$$

Proof

Step	Hyp	Ref	Expression
1		eleqtrd.1	$$⊢\phi \to A\in B$$
2		eleqtrd.2	$$⊢\phi \to B=C$$
3	2	eleq2d	$$⊢\phi \to \left(A\in B\leftrightarrow A\in C\right)$$
4	1 3	mpbid	$$⊢\phi \to A\in C$$