Theorem neneqd

Description: Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011)

	Ref	Expression
Hypothesis	neneqd.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
Assertion	neneqd	⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		neneqd.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
2		df-ne	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
3	1 2	sylib	⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 )