Theorem neqned 2252

Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2266. One-way deduction form of df-ne 2246. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2295. (Revised by Wolf Lammen, 22-Nov-2019.)

Hypothesis

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

Assertion

Ref	Expression
neqned	⊢ (𝜑 → 𝐴 ≠ 𝐵)

Proof of Theorem neqned

Step	Hyp	Ref	Expression
1		neqned.1	. 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
2		df-ne 2246	. 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)
3	1, 2	sylibr 132	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1284   ≠ wne 2245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115  df-ne 2246

This theorem is referenced by:  neqne  2253  flqltnz  9289  bezoutlemle  10397  eucalgval2  10435  eucalglt  10439  lcmval  10445  lcmcllem  10449  isprm2  10499  sqne2sq  10555