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Theorem nbbndc 1328
Description: Move negation outside of biconditional, for decidable propositions. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)
Assertion
Ref Expression
nbbndc (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓))))

Proof of Theorem nbbndc
StepHypRef Expression
1 xor3dc 1321 . . . . 5 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))
21imp 122 . . . 4 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))
3 con2bidc 805 . . . . 5 (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))))
43imp 122 . . . 4 ((DECID 𝜑DECID 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))
52, 4bitrd 186 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))
6 bicom 138 . . 3 ((𝜓 ↔ ¬ 𝜑) ↔ (¬ 𝜑𝜓))
75, 6syl6rbb 195 . 2 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓)))
87ex 113 1 (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓))))
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:  biassdc  1329
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