Theorem eqeltrid

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

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

Proof

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