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Theorem necon4ai 2808
Description: Contrapositive inference for inequality. (Contributed by NM, 16-Jan-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 22-Nov-2019.)
Hypothesis
Ref Expression
necon4ai.1 (𝐴𝐵 → ¬ 𝜑)
Assertion
Ref Expression
necon4ai (𝜑𝐴 = 𝐵)

Proof of Theorem necon4ai
StepHypRef Expression
1 notnot 134 . 2 (𝜑 → ¬ ¬ 𝜑)
2 necon4ai.1 . . 3 (𝐴𝐵 → ¬ 𝜑)
32necon1bi 2805 . 2 (¬ ¬ 𝜑𝐴 = 𝐵)
41, 3syl 17 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1474  wne 2775
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 2777
This theorem is referenced by:  necon4i  2812  dmsn0el  5504  cfeq0  8934  funsneqopb  40147
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