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Theorem annimdc 845
 Description: Express conjunction in terms of implication. The forward direction, annimim 782, is valid for all propositions, but as an equivalence, it requires a decidability condition. (Contributed by Jim Kingdon, 25-Apr-2018.)
Assertion
Ref Expression
annimdc (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))))

Proof of Theorem annimdc
StepHypRef Expression
1 imandc 786 . . . 4 (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
21adantl 262 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
3 dcim 784 . . . . 5 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
43imp 115 . . . 4 ((DECID 𝜑DECID 𝜓) → DECID (𝜑𝜓))
5 dcn 746 . . . . . 6 (DECID 𝜓DECID ¬ 𝜓)
6 dcan 842 . . . . . 6 (DECID 𝜑 → (DECID ¬ 𝜓DECID (𝜑 ∧ ¬ 𝜓)))
75, 6syl5 28 . . . . 5 (DECID 𝜑 → (DECID 𝜓DECID (𝜑 ∧ ¬ 𝜓)))
87imp 115 . . . 4 ((DECID 𝜑DECID 𝜓) → DECID (𝜑 ∧ ¬ 𝜓))
9 con2bidc 769 . . . 4 (DECID (𝜑𝜓) → (DECID (𝜑 ∧ ¬ 𝜓) → (((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓)))))
104, 8, 9sylc 56 . . 3 ((DECID 𝜑DECID 𝜓) → (((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))))
112, 10mpbid 135 . 2 ((DECID 𝜑DECID 𝜓) → ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓)))
1211ex 108 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (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:  xordidc  1290
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