Theorem eqeltrdi

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

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

Proof

Step	Hyp	Ref	Expression
1		eqeltrdi.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eqeltrdi.2	⊢ 𝐵 ∈ 𝐶
3	2	a1i	⊢ ( 𝜑 → 𝐵 ∈ 𝐶 )
4	1 3	eqeltrd	⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )