Theorem eleq1i

Description: Inference from equality to equivalence of membership. (Contributed by NM, 21-Jun-1993)

	Ref	Expression
Hypothesis	eleq1i.1	⊢ 𝐴 = 𝐵
Assertion	eleq1i	⊢ ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eleq1i.1	⊢ 𝐴 = 𝐵
2		eleq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 ) )
3	1 2	ax-mp	⊢ ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 )