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Theorem nonconne 2983
Description: Law of noncontradiction with equality and inequality. (Contributed by NM, 3-Feb-2012.) (Proof shortened by Wolf Lammen, 21-Dec-2019.)
Assertion
Ref Expression
nonconne ¬ (𝐴 = 𝐵𝐴𝐵)

Proof of Theorem nonconne
StepHypRef Expression
1 fal 1668 . 2 ¬ ⊥
2 eqneqall 2982 . . 3 (𝐴 = 𝐵 → (𝐴𝐵 → ⊥))
32imp 396 . 2 ((𝐴 = 𝐵𝐴𝐵) → ⊥)
41, 3mto 189 1 ¬ (𝐴 = 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 385   = wceq 1653  wfal 1666  wne 2971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 199  df-an 386  df-tru 1657  df-fal 1667  df-ne 2972
This theorem is referenced by:  osumcllem11N  35987  pexmidlem8N  35998  dochexmidlem8  37488
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