Theorem dcbi 938

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 842	. . 3 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 → 𝜓)))
2		dcim 842	. . . 4 ⊢ (DECID 𝜓 → (DECID 𝜑 → DECID (𝜓 → 𝜑)))
3	2	com12 30	. . 3 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜓 → 𝜑)))
4		dcan 935	. . . 4 ⊢ ((DECID (𝜑 → 𝜓) ∧ DECID (𝜓 → 𝜑)) → DECID ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
5	4	ex 115	. . 3 ⊢ (DECID (𝜑 → 𝜓) → (DECID (𝜓 → 𝜑) → DECID ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))))
6	1, 3, 5	syl6c 66	. 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))))
7		dfbi2 388	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
8	7	dcbii 841	. 2 ⊢ (DECID (𝜑 ↔ 𝜓) ↔ DECID ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
9	6, 8	imbitrrdi 162	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ↔ 𝜓)))

Syntax hints:   → 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-dc 836

This theorem is referenced by:  xor3dc  1398  pm5.15dc  1400  bilukdc  1407  xordidc  1410