Theorem eleqtrrd

Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004)

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

Proof

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