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Theorem necon1abid 2980
Description: Contrapositive deduction for inequality. (Contributed by NM, 21-Aug-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
Hypothesis
Ref Expression
necon1abid.1 (𝜑 → (¬ 𝜓𝐴 = 𝐵))
Assertion
Ref Expression
necon1abid (𝜑 → (𝐴𝐵𝜓))

Proof of Theorem necon1abid
StepHypRef Expression
1 notnotb 304 . 2 (𝜓 ↔ ¬ ¬ 𝜓)
2 necon1abid.1 . . 3 (𝜑 → (¬ 𝜓𝐴 = 𝐵))
32necon3bbid 2979 . 2 (𝜑 → (¬ ¬ 𝜓𝐴𝐵))
41, 3syl5rbb 273 1 (𝜑 → (𝐴𝐵𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1630  wne 2942
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 197  df-ne 2943
This theorem is referenced by:  lttri2  10321  xrlttri2  12179  ioon0  12405  lssne0  19160  xmetgt0  22382  sotrine  31990
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