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Theorem necon1abiddc 2317
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon1abiddc.1 (𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴 = 𝐵)))
Assertion
Ref Expression
necon1abiddc (𝜑 → (DECID 𝜓 → (𝐴𝐵𝜓)))

Proof of Theorem necon1abiddc
StepHypRef Expression
1 necon1abiddc.1 . . 3 (𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴 = 𝐵)))
21con1biddc 808 . 2 (𝜑 → (DECID 𝜓 → (¬ 𝐴 = 𝐵𝜓)))
3 df-ne 2256 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
43bibi1i 226 . 2 ((𝐴𝐵𝜓) ↔ (¬ 𝐴 = 𝐵𝜓))
52, 4syl6ibr 160 1 (𝜑 → (DECID 𝜓 → (𝐴𝐵𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  DECID wdc 780   = wceq 1289  wne 2255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781  df-ne 2256
This theorem is referenced by:  necon2abiddc  2321
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