Theorem dcbi 877

Description: An equivalence of two decidable propositions is decidable. (Contributed by Jim Kingdon, 12-Apr-2018.)

Assertion

Ref	Expression
dcbi	⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ↔ 𝜓)))

Proof of Theorem dcbi

Step	Hyp	Ref	Expression
1		dcim 817	. . 3 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 → 𝜓)))
2		dcim 817	. . . 4 ⊢ (DECID 𝜓 → (DECID 𝜑 → DECID (𝜓 → 𝜑)))
3	2	com12 30	. . 3 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜓 → 𝜑)))
4		dcan 875	. . 3 ⊢ (DECID (𝜑 → 𝜓) → (DECID (𝜓 → 𝜑) → DECID ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))))
5	1, 3, 4	syl6c 65	. 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))))
6		dfbi2 380	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
7	6	dcbii 780	. 2 ⊢ (DECID (𝜑 ↔ 𝜓) ↔ DECID ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
8	5, 7	syl6ibr 160	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ↔ 𝜓)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  DECID wdc 775

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 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  xor3dc  1318  pm5.15dc  1320  bilukdc  1327  xordidc  1330