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Theorem necon4bbiddc 2279
 Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
Hypothesis
Ref Expression
necon4bbiddc.1 (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (¬ 𝜓𝐴𝐵))))
Assertion
Ref Expression
necon4bbiddc (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝜓𝐴 = 𝐵))))

Proof of Theorem necon4bbiddc
StepHypRef Expression
1 necon4bbiddc.1 . . . . . 6 (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (¬ 𝜓𝐴𝐵))))
2 bicom 128 . . . . . 6 ((¬ 𝜓𝐴𝐵) ↔ (𝐴𝐵 ↔ ¬ 𝜓))
31, 2syl8ib 155 . . . . 5 (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝐴𝐵 ↔ ¬ 𝜓))))
43com23 72 . . . 4 (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴𝐵 ↔ ¬ 𝜓))))
54necon4abiddc 2278 . . 3 (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 = 𝐵𝜓))))
65com23 72 . 2 (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵𝜓))))
7 bicom 128 . 2 ((𝐴 = 𝐵𝜓) ↔ (𝜓𝐴 = 𝐵))
86, 7syl8ib 155 1 (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝜓𝐴 = 𝐵))))
 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-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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