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Theorem necon2bbiidc 2392
 Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2bbii.1 (DECID 𝐴 = 𝐵 → (𝜑𝐴𝐵))
Assertion
Ref Expression
necon2bbiidc (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜑))

Proof of Theorem necon2bbiidc
StepHypRef Expression
1 necon2bbii.1 . . . 4 (DECID 𝐴 = 𝐵 → (𝜑𝐴𝐵))
21bicomd 140 . . 3 (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜑))
32necon1bbiidc 2388 . 2 (DECID 𝐴 = 𝐵 → (¬ 𝜑𝐴 = 𝐵))
43bicomd 140 1 (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104  DECID wdc 820   = wceq 1335   ≠ wne 2327 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699 This theorem depends on definitions:  df-bi 116  df-dc 821  df-ne 2328 This theorem is referenced by: (None)
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