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Theorem neqned 2256
Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2270. One-way deduction form of df-ne 2250. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2299. (Revised by Wolf Lammen, 22-Nov-2019.)
Hypothesis
Ref Expression
neqned.1 (𝜑 → ¬ 𝐴 = 𝐵)
Assertion
Ref Expression
neqned (𝜑𝐴𝐵)

Proof of Theorem neqned
StepHypRef Expression
1 neqned.1 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
2 df-ne 2250 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
31, 2sylibr 132 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1285  wne 2249
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 2250
This theorem is referenced by:  neqne  2257  tfr1onlemsucaccv  6038  tfrcllemsucaccv  6051  djune  6676  flqltnz  9583  bezoutlemle  10777  eucalgval2  10815  eucalglt  10819  lcmval  10825  lcmcllem  10829  isprm2  10879  sqne2sq  10935
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