Theorem eleqtrid

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

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

Proof

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