Theorem neleq2

Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994) (Proof shortened by Wolf Lammen, 25-Nov-2019)

	Ref	Expression
Assertion	neleq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eqidd	⊢ ( 𝐴 = 𝐵 → 𝐶 = 𝐶 )
2		id	⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 )
3	1 2	neleq12d	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ∉ 𝐴 ↔ 𝐶 ∉ 𝐵 ) )