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Theorem nonconne 3023
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 1552 . 2 ¬ ⊥
2 eqneqall 3022 . . 3 (𝐴 = 𝐵 → (𝐴𝐵 → ⊥))
32imp 410 . 2 ((𝐴 = 𝐵𝐴𝐵) → ⊥)
41, 3mto 200 1 ¬ (𝐴 = 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1538  wfal 1550  wne 3011
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 210  df-an 400  df-tru 1541  df-fal 1551  df-ne 3012
This theorem is referenced by:  osumcllem11N  37220  pexmidlem8N  37231  dochexmidlem8  38721
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