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Theorem necon4bbid 2822
Description: Contrapositive law deduction for inequality. (Contributed by NM, 9-May-2012.)
Hypothesis
Ref Expression
necon4bbid.1 (𝜑 → (¬ 𝜓𝐴𝐵))
Assertion
Ref Expression
necon4bbid (𝜑 → (𝜓𝐴 = 𝐵))

Proof of Theorem necon4bbid
StepHypRef Expression
1 necon4bbid.1 . . . 4 (𝜑 → (¬ 𝜓𝐴𝐵))
21bicomd 211 . . 3 (𝜑 → (𝐴𝐵 ↔ ¬ 𝜓))
32necon4abid 2821 . 2 (𝜑 → (𝐴 = 𝐵𝜓))
43bicomd 211 1 (𝜑 → (𝜓𝐴 = 𝐵))
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:  fzn  12185  lgsqr  24820
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