Theorem 3eltr4d

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

	Ref	Expression
Hypotheses	3eltr4d.1	$$⊢\phi \to A\in B$$
	3eltr4d.2	$$⊢\phi \to C=A$$
	3eltr4d.3	$$⊢\phi \to D=B$$
Assertion	3eltr4d	$$⊢\phi \to C\in D$$

Proof

Step	Hyp	Ref	Expression
1		3eltr4d.1	$$⊢\phi \to A\in B$$
2		3eltr4d.2	$$⊢\phi \to C=A$$
3		3eltr4d.3	$$⊢\phi \to D=B$$
4	1 3	eleqtrrd	$$⊢\phi \to A\in D$$
5	2 4	eqeltrd	$$⊢\phi \to C\in D$$