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Metamath Proof Explorer

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 )