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Theorem necon4abiddc 2319
Description: Contrapositive law deduction for inequality. (Contributed by Jim Kingdon, 18-May-2018.)
Hypothesis
Ref Expression
necon4abiddc.1 (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴𝐵 ↔ ¬ 𝜓))))
Assertion
Ref Expression
necon4abiddc (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 = 𝐵𝜓))))

Proof of Theorem necon4abiddc
StepHypRef Expression
1 necon4abiddc.1 . . 3 (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴𝐵 ↔ ¬ 𝜓))))
2 df-ne 2247 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
32bibi1i 226 . . 3 ((𝐴𝐵 ↔ ¬ 𝜓) ↔ (¬ 𝐴 = 𝐵 ↔ ¬ 𝜓))
41, 3syl8ib 164 . 2 (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝜓))))
54con4biddc 788 1 (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 = 𝐵𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  DECID wdc 776   = wceq 1285  wne 2246
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-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777  df-ne 2247
This theorem is referenced by:  necon4bbiddc  2320  necon4biddc  2321
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