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Theorem necon2abiddc 2271
 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 128 . . . 4 ((𝐴 = 𝐵 ↔ ¬ 𝜓) ↔ (¬ 𝜓𝐴 = 𝐵))
31, 2syl6ib 150 . . 3 (𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴 = 𝐵)))
43necon1abiddc 2267 . 2 (𝜑 → (DECID 𝜓 → (𝐴𝐵𝜓)))
5 bicom 128 . 2 ((𝐴𝐵𝜓) ↔ (𝜓𝐴𝐵))
64, 5syl6ib 150 1 (𝜑 → (DECID 𝜓 → (𝜓𝐴𝐵)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  DECID wdc 742   = wceq 1243   ≠ wne 2204 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743  df-ne 2206 This theorem is referenced by: (None)
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