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Theorem nbbndc 1285
 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 1278 . . . . 5 (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓))))
21imp 115 . . . 4 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ (𝜑 ↔ ¬ 𝜓)))
3 con2bidc 769 . . . . 5 (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑))))
43imp 115 . . . 4 ((DECID 𝜑DECID 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))
52, 4bitrd 177 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (𝜑𝜓) ↔ (𝜓 ↔ ¬ 𝜑)))
6 bicom 128 . . 3 ((𝜓 ↔ ¬ 𝜑) ↔ (¬ 𝜑𝜓))
75, 6syl6rbb 186 . 2 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓)))
87ex 108 1 (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ ¬ (𝜑𝜓))))
 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:  biassdc  1286
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