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Theorem necon2abiddc 2400
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2abiddc.1 (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))
Assertion
Ref Expression
necon2abiddc (𝜑 → (DECID 𝜓 → (𝜓𝐴𝐵)))

Proof of Theorem necon2abiddc
StepHypRef Expression
1 necon2abiddc.1 . . . 4 (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))
2 bicom 139 . . . 4 ((𝐴 = 𝐵 ↔ ¬ 𝜓) ↔ (¬ 𝜓𝐴 = 𝐵))
31, 2syl6ib 160 . . 3 (𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴 = 𝐵)))
43necon1abiddc 2396 . 2 (𝜑 → (DECID 𝜓 → (𝐴𝐵𝜓)))
5 bicom 139 . 2 ((𝐴𝐵𝜓) ↔ (𝜓𝐴𝐵))
64, 5syl6ib 160 1 (𝜑 → (DECID 𝜓 → (𝜓𝐴𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  DECID wdc 824   = wceq 1342  wne 2334
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-stab 821  df-dc 825  df-ne 2335
This theorem is referenced by: (None)
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