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Theorem con2bidc 876
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 875 . . . . 5 (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))))
21imp 124 . . . 4 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑)))
3 bicom 140 . . . 4 ((¬ 𝜑𝜓) ↔ (𝜓 ↔ ¬ 𝜑))
4 bicom 140 . . . 4 ((¬ 𝜓𝜑) ↔ (𝜑 ↔ ¬ 𝜓))
52, 3, 43bitr3g 222 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜓 ↔ ¬ 𝜑) ↔ (𝜑 ↔ ¬ 𝜓)))
65bicomd 141 . 2 ((DECID 𝜑DECID 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))
76ex 115 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  DECID wdc 835
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-stab 832  df-dc 836
This theorem is referenced by:  annimdc  938  pm4.55dc  939  orandc  940  nbbndc  1404
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