Theorem eleq2

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

	Ref	Expression
Assertion	eleq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 )
2	1	eleq2d	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵 ) )