Theorem eqeltrrdi

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

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

Proof

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