Theorem 3eltr3g

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	3eltr3g.1	$$⊢\phi \to A\in B$$
	3eltr3g.2	$$⊢A=C$$
	3eltr3g.3	$$⊢B=D$$
Assertion	3eltr3g	$$⊢\phi \to C\in D$$

Proof

Step	Hyp	Ref	Expression
1		3eltr3g.1	$$⊢\phi \to A\in B$$
2		3eltr3g.2	$$⊢A=C$$
3		3eltr3g.3	$$⊢B=D$$
4	2 1	eqeltrrid	$$⊢\phi \to C\in B$$
5	4 3	eleqtrdi	$$⊢\phi \to C\in D$$