Theorem dcbi 926

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 831	. . 3 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 → 𝜓)))
2		dcim 831	. . . 4 ⊢ (DECID 𝜓 → (DECID 𝜑 → DECID (𝜓 → 𝜑)))
3	2	com12 30	. . 3 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜓 → 𝜑)))
4		dcan2 924	. . 3 ⊢ (DECID (𝜑 → 𝜓) → (DECID (𝜓 → 𝜑) → DECID ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))))
5	1, 3, 4	syl6c 66	. 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))))
6		dfbi2 386	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
7	6	dcbii 830	. 2 ⊢ (DECID (𝜑 ↔ 𝜓) ↔ DECID ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
8	5, 7	syl6ibr 161	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ↔ 𝜓)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-dc 825

This theorem is referenced by:  xor3dc  1377  pm5.15dc  1379  bilukdc  1386  xordidc  1389