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Theorem necon2bbiddc 2414
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2bbiddc.1 (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓𝐴𝐵)))
Assertion
Ref Expression
necon2bbiddc (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))

Proof of Theorem necon2bbiddc
StepHypRef Expression
1 necon2bbiddc.1 . . . 4 (𝜑 → (DECID 𝐴 = 𝐵 → (𝜓𝐴𝐵)))
2 bicom 140 . . . 4 ((𝜓𝐴𝐵) ↔ (𝐴𝐵𝜓))
31, 2imbitrdi 161 . . 3 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜓)))
43necon1bbiddc 2410 . 2 (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓𝐴 = 𝐵)))
5 bicom 140 . 2 ((¬ 𝜓𝐴 = 𝐵) ↔ (𝐴 = 𝐵 ↔ ¬ 𝜓))
64, 5imbitrdi 161 1 (𝜑 → (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  DECID wdc 834   = wceq 1353  wne 2347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-ne 2348
This theorem is referenced by: (None)
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