Theorem 3eltr3d

Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017)

	Ref	Expression
Hypotheses	3eltr3d.1	$$⊢\phi \to A\in B$$
	3eltr3d.2	$$⊢\phi \to A=C$$
	3eltr3d.3	$$⊢\phi \to B=D$$
Assertion	3eltr3d	$$⊢\phi \to C\in D$$

Proof

Step	Hyp	Ref	Expression
1		3eltr3d.1	$$⊢\phi \to A\in B$$
2		3eltr3d.2	$$⊢\phi \to A=C$$
3		3eltr3d.3	$$⊢\phi \to B=D$$
4	1 3	eleqtrd	$$⊢\phi \to A\in D$$
5	2 4	eqeltrrd	$$⊢\phi \to C\in D$$