Theorem eleqtrdi

Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006)

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

Proof

Step	Hyp	Ref	Expression
1		eleqtrdi.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2		eleqtrdi.2	⊢ 𝐵 = 𝐶
3	2	a1i	⊢ ( 𝜑 → 𝐵 = 𝐶 )
4	1 3	eleqtrd	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )