Theorem neqned

Description: If it is not the case that two classes are equal, then they are unequal. Converse of neneqd . One-way deduction form of df-ne . (Contributed by David Moews, 28-Feb-2017) Allow a shortening of necon3bi . (Revised by Wolf Lammen, 22-Nov-2019)

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

Proof

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