Theorem 3eltr4g

Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017) (Proof shortened by Wolf Lammen, 23-Nov-2019)

	Ref	Expression
Hypotheses	3eltr4g.1	$$⊢\phi \to A\in B$$
	3eltr4g.2	$$⊢C=A$$
	3eltr4g.3	$$⊢D=B$$
Assertion	3eltr4g	$$⊢\phi \to C\in D$$

Proof

Step	Hyp	Ref	Expression
1		3eltr4g.1	$$⊢\phi \to A\in B$$
2		3eltr4g.2	$$⊢C=A$$
3		3eltr4g.3	$$⊢D=B$$
4	2 1	eqeltrid	$$⊢\phi \to C\in B$$
5	4 3	eleqtrrdi	$$⊢\phi \to C\in D$$