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Theorem con2bidc 805
Description: Contraposition. (Contributed by Jim Kingdon, 17-Apr-2018.)
Assertion
Ref Expression
con2bidc (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))))

Proof of Theorem con2bidc
StepHypRef Expression
1 con1bidc 804 . . . . 5 (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))))
21imp 122 . . . 4 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑)))
3 bicom 138 . . . 4 ((¬ 𝜑𝜓) ↔ (𝜓 ↔ ¬ 𝜑))
4 bicom 138 . . . 4 ((¬ 𝜓𝜑) ↔ (𝜑 ↔ ¬ 𝜓))
52, 3, 43bitr3g 220 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜓 ↔ ¬ 𝜑) ↔ (𝜑 ↔ ¬ 𝜓)))
65bicomd 139 . 2 ((DECID 𝜑DECID 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))
76ex 113 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))))
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:  annimdc  881  pm4.55dc  882  orandc  883  nbbndc  1328
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