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Theorem necon2abid 2983
Description: Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.) (Proof shortened by Wolf Lammen, 24-Nov-2019.)
Hypothesis
Ref Expression
necon2abid.1 (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))
Assertion
Ref Expression
necon2abid (𝜑 → (𝜓𝐴𝐵))

Proof of Theorem necon2abid
StepHypRef Expression
1 notnotb 314 . 2 (𝜓 ↔ ¬ ¬ 𝜓)
2 necon2abid.1 . . 3 (𝜑 → (𝐴 = 𝐵 ↔ ¬ 𝜓))
32necon3abid 2977 . 2 (𝜑 → (𝐴𝐵 ↔ ¬ ¬ 𝜓))
41, 3bitr4id 289 1 (𝜑 → (𝜓𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205   = wceq 1541  wne 2940
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 206  df-ne 2941
This theorem is referenced by:  sossfld  6185  funeldmb  7358  fin23lem24  10319  isf32lem4  10353  sqgt0sr  11103  leltne  11305  xrleltne  13126  xrltne  13144  ge0nemnf  13154  xlt2add  13241  supxrbnd  13309  supxrre2  13312  ioopnfsup  13831  icopnfsup  13832  xblpnfps  23908  xblpnf  23909  nmoreltpnf  30060  nmopreltpnf  31160  elprneb  45818
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