Theorem eleqtrrdi

Description: A membership and equality inference. (Contributed by NM, 24-Apr-2005)

	Ref	Expression
Hypotheses	eleqtrrdi.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
	eleqtrrdi.2	⊢ 𝐶 = 𝐵
Assertion	eleqtrrdi	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eleqtrrdi.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2		eleqtrrdi.2	⊢ 𝐶 = 𝐵
3	2	eqcomi	⊢ 𝐵 = 𝐶
4	1 3	eleqtrdi	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )