Theorem eleqtrd

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

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

Proof

Step	Hyp	Ref	Expression
1		eleqtrd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2		eleqtrd.2	⊢ ( 𝜑 → 𝐵 = 𝐶 )
3	2	eleq2d	⊢ ( 𝜑 → ( 𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶 ) )
4	1 3	mpbid	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )