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	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
	3eltr3g.2	⊢ 𝐴 = 𝐶
	3eltr3g.3	⊢ 𝐵 = 𝐷
Assertion	3eltr3g	⊢ ( 𝜑 → 𝐶 ∈ 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		3eltr3g.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2		3eltr3g.2	⊢ 𝐴 = 𝐶
3		3eltr3g.3	⊢ 𝐵 = 𝐷
4	2 1	eqeltrrid	⊢ ( 𝜑 → 𝐶 ∈ 𝐵 )
5	4 3	eleqtrdi	⊢ ( 𝜑 → 𝐶 ∈ 𝐷 )