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Theorem necon2abiddc 2286
 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 132 . . . 4 ((𝐴 = 𝐵 ↔ ¬ 𝜓) ↔ (¬ 𝜓𝐴 = 𝐵))
31, 2syl6ib 154 . . 3 (𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴 = 𝐵)))
43necon1abiddc 2282 . 2 (𝜑 → (DECID 𝜓 → (𝐴𝐵𝜓)))
5 bicom 132 . 2 ((𝐴𝐵𝜓) ↔ (𝜓𝐴𝐵))
64, 5syl6ib 154 1 (𝜑 → (DECID 𝜓 → (𝜓𝐴𝐵)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 102  DECID wdc 753   = wceq 1259   ≠ wne 2220 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640 This theorem depends on definitions:  df-bi 114  df-dc 754  df-ne 2221 This theorem is referenced by: (None)
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