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Theorem necon1bbiidc 2418
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon1bbiidc.1 (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜑))
Assertion
Ref Expression
necon1bbiidc (DECID 𝐴 = 𝐵 → (¬ 𝜑𝐴 = 𝐵))

Proof of Theorem necon1bbiidc
StepHypRef Expression
1 df-ne 2358 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon1bbiidc.1 . . 3 (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜑))
31, 2bitr3id 194 . 2 (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵𝜑))
43con1biidc 878 1 (DECID 𝐴 = 𝐵 → (¬ 𝜑𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  DECID wdc 835   = wceq 1363  wne 2357
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 615  ax-in2 616  ax-io 710
This theorem depends on definitions:  df-bi 117  df-dc 836  df-ne 2358
This theorem is referenced by:  necon2bbiidc  2422
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