Theorem annimdc 939

Description: Express conjunction in terms of implication. The forward direction, annimim 687, 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) ↔ ¬ (𝜑 → 𝜓))))

Proof of Theorem annimdc

Step	Hyp	Ref	Expression
1		imandc 890	. . . 4 ⊢ (DECID 𝜓 → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
2	1	adantl 277	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)))
3		dcim 842	. . . . 5 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 → 𝜓)))
4	3	imp 124	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 → 𝜓))
5		dcn 843	. . . . 5 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
6		dcan 935	. . . . 5 ⊢ ((DECID 𝜑 ∧ DECID ¬ 𝜓) → DECID (𝜑 ∧ ¬ 𝜓))
7	5, 6	sylan2 286	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∧ ¬ 𝜓))
8		con2bidc 876	. . . 4 ⊢ (DECID (𝜑 → 𝜓) → (DECID (𝜑 ∧ ¬ 𝜓) → (((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓)))))
9	4, 7, 8	sylc 62	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (((𝜑 → 𝜓) ↔ ¬ (𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))))
10	2, 9	mpbid 147	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓)))
11	10	ex 115	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836

This theorem is referenced by:  xordidc  1410