Theorem eleq1

Description: Equality implies equivalence of membership. (Contributed by NM, 26-May-1993) (Proof shortened by Wolf Lammen, 20-Nov-2019)

	Ref	Expression
Assertion	eleq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 )
2	1	eleq1d	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 ) )