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Theorem con2bidc 769
 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 768 . . . . 5 (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑))))
21imp 115 . . . 4 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) ↔ (¬ 𝜓𝜑)))
3 bicom 128 . . . 4 ((¬ 𝜑𝜓) ↔ (𝜓 ↔ ¬ 𝜑))
4 bicom 128 . . . 4 ((¬ 𝜓𝜑) ↔ (𝜑 ↔ ¬ 𝜓))
52, 3, 43bitr3g 211 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜓 ↔ ¬ 𝜑) ↔ (𝜑 ↔ ¬ 𝜓)))
65bicomd 129 . 2 ((DECID 𝜑DECID 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))
76ex 108 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  DECID wdc 742 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  annimdc  845  pm4.55dc  846  nbbndc  1285
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