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Theorem necon4ai 2980
 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 144 . 2 (𝜑 → ¬ ¬ 𝜑)
2 necon4ai.1 . . 3 (𝐴𝐵 → ¬ 𝜑)
32necon1bi 2977 . 2 (¬ ¬ 𝜑𝐴 = 𝐵)
41, 3syl 17 1 (𝜑𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ≠ wne 2949 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 210  df-ne 2950 This theorem is referenced by:  necon4i  2984  dmsn0el  6033  funsneqopb  6898  cfeq0  9701
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