Theorem neeq1

Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994) (Proof shortened by Wolf Lammen, 18-Nov-2019)

	Ref	Expression
Assertion	neeq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐴 = 𝐵 → 𝐴 = 𝐵 )
2	1	neeq1d	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶 ) )