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Theorem con1bidc 804
Description: Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
Assertion
Ref Expression
con1bidc (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))))

Proof of Theorem con1bidc
StepHypRef Expression
1 con1biimdc 803 . . . 4 (DECID 𝜑 → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
21adantr 270 . . 3 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
3 con1biimdc 803 . . . 4 (DECID 𝜓 → ((¬ 𝜓𝜑) → (¬ 𝜑𝜓)))
43adantl 271 . . 3 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜓𝜑) → (¬ 𝜑𝜓)))
52, 4impbid 127 . 2 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑)))
65ex 113 1 (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  DECID wdc 778
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 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 779
This theorem is referenced by:  con2bidc  805
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