Theorem eleqtrrid

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

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

Proof

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