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Theorem annimdc 881
Description: Express conjunction in terms of implication. The forward direction, annimim 818, 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 822 . . . 4 (DECID 𝜓 → ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
21adantl 271 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
3 dcim 820 . . . . 5 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
43imp 122 . . . 4 ((DECID 𝜑DECID 𝜓) → DECID (𝜑𝜓))
5 dcn 782 . . . . . 6 (DECID 𝜓DECID ¬ 𝜓)
6 dcan 878 . . . . . 6 (DECID 𝜑 → (DECID ¬ 𝜓DECID (𝜑 ∧ ¬ 𝜓)))
75, 6syl5 32 . . . . 5 (DECID 𝜑 → (DECID 𝜓DECID (𝜑 ∧ ¬ 𝜓)))
87imp 122 . . . 4 ((DECID 𝜑DECID 𝜓) → DECID (𝜑 ∧ ¬ 𝜓))
9 con2bidc 805 . . . 4 (DECID (𝜑𝜓) → (DECID (𝜑 ∧ ¬ 𝜓) → (((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓)))))
104, 8, 9sylc 61 . . 3 ((DECID 𝜑DECID 𝜓) → (((𝜑𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))))
112, 10mpbid 145 . 2 ((DECID 𝜑DECID 𝜓) → ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓)))
1211ex 113 1 (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (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:  xordidc  1333
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