Theorem eqeltrrd

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

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

Proof

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