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Theorem necon4abid 2821
Description: Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
Hypothesis
Ref Expression
necon4abid.1 (𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))
Assertion
Ref Expression
necon4abid (𝜑 → (𝐴 = 𝐵𝜓))

Proof of Theorem necon4abid
StepHypRef Expression
1 notnotb 302 . 2 (𝜓 ↔ ¬ ¬ 𝜓)
2 necon4abid.1 . . 3 (𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))
32necon1bbid 2820 . 2 (𝜑 → (¬ ¬ 𝜓𝐴 = 𝐵))
41, 3syl5rbb 271 1 (𝜑 → (𝐴 = 𝐵𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194   = wceq 1474  wne 2779
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 195  df-ne 2781
This theorem is referenced by:  necon4bbid  2822  necon2bbid  2824  birthdaylem3  24397  lgsprme0  24781  nmounbi  26821
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