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Theorem necon4bbiddc 2294
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 132 . . . . . 6 ((¬ 𝜓𝐴𝐵) ↔ (𝐴𝐵 ↔ ¬ 𝜓))
31, 2syl8ib 159 . . . . 5 (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝐴𝐵 ↔ ¬ 𝜓))))
43com23 76 . . . 4 (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴𝐵 ↔ ¬ 𝜓))))
54necon4abiddc 2293 . . 3 (𝜑 → (DECID 𝐴 = 𝐵 → (DECID 𝜓 → (𝐴 = 𝐵𝜓))))
65com23 76 . 2 (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵𝜓))))
7 bicom 132 . 2 ((𝐴 = 𝐵𝜓) ↔ (𝜓𝐴 = 𝐵))
86, 7syl8ib 159 1 (𝜑 → (DECID 𝜓 → (DECID 𝐴 = 𝐵 → (𝜓𝐴 = 𝐵))))
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-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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