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Metamath Proof Explorer

Theorem neleq1

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

		Ref	Expression
	Assertion	neleq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶 ) )

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