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Theorem con1biimdc 778
Description: Contraposition. (Contributed by Jim Kingdon, 4-Apr-2018.)
Assertion
Ref Expression
con1biimdc (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))

Proof of Theorem con1biimdc
StepHypRef Expression
1 bi1 115 . . 3 ((¬ 𝜑𝜓) → (¬ 𝜑𝜓))
2 con1dc 764 . . 3 (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
31, 2syl5 32 . 2 (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
4 bi2 125 . . . 4 ((¬ 𝜑𝜓) → (𝜓 → ¬ 𝜑))
54con2d 564 . . 3 ((¬ 𝜑𝜓) → (𝜑 → ¬ 𝜓))
65a1i 9 . 2 (DECID 𝜑 → ((¬ 𝜑𝜓) → (𝜑 → ¬ 𝜓)))
73, 6impbidd 122 1 (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  DECID wdc 753
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-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by:  con1bidc  779  con1biddc  781
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