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Theorem necon1bbiidc 2316
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 2256 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon1bbiidc.1 . . 3 (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜑))
31, 2syl5bbr 192 . 2 (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵𝜑))
43con1biidc 809 1 (DECID 𝐴 = 𝐵 → (¬ 𝜑𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  DECID wdc 780   = wceq 1289  wne 2255
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-in1 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781  df-ne 2256
This theorem is referenced by:  necon2bbii  2320
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