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Theorem necon1abid 3044
 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 317 . 2 (𝜓 ↔ ¬ ¬ 𝜓)
2 necon1abid.1 . . 3 (𝜑 → (¬ 𝜓𝐴 = 𝐵))
32necon3bbid 3043 . 2 (𝜑 → (¬ ¬ 𝜓𝐴𝐵))
41, 3syl5rbb 286 1 (𝜑 → (𝐴𝐵𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1537   ≠ wne 3006 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 209  df-ne 3007 This theorem is referenced by:  lttri2  10701  xrlttri2  12514  ioon0  12743  lssne0  19698  xmetgt0  22944  sotrine  33011
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