Theorem dcim 784

Description: An implication between two decidable propositions is decidable. (Contributed by Jim Kingdon, 28-Mar-2018.)

Assertion

Ref	Expression
dcim	⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 → 𝜓)))

Proof of Theorem dcim

Step	Hyp	Ref	Expression
1		df-dc 743	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		df-dc 743	. . . . . . . 8 ⊢ (DECID 𝜓 ↔ (𝜓 ∨ ¬ 𝜓))
3	2	anbi2i 430	. . . . . . 7 ⊢ ((𝜑 ∧ DECID 𝜓) ↔ (𝜑 ∧ (𝜓 ∨ ¬ 𝜓)))
4		andi 731	. . . . . . 7 ⊢ ((𝜑 ∧ (𝜓 ∨ ¬ 𝜓)) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓)))
5	3, 4	bitri 173	. . . . . 6 ⊢ ((𝜑 ∧ DECID 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓)))
6		pm3.4 316	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 → 𝜓))
7		annimim 782	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
8	6, 7	orim12i 676	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓)) → ((𝜑 → 𝜓) ∨ ¬ (𝜑 → 𝜓)))
9	5, 8	sylbi 114	. . . . 5 ⊢ ((𝜑 ∧ DECID 𝜓) → ((𝜑 → 𝜓) ∨ ¬ (𝜑 → 𝜓)))
10		df-dc 743	. . . . 5 ⊢ (DECID (𝜑 → 𝜓) ↔ ((𝜑 → 𝜓) ∨ ¬ (𝜑 → 𝜓)))
11	9, 10	sylibr 137	. . . 4 ⊢ ((𝜑 ∧ DECID 𝜓) → DECID (𝜑 → 𝜓))
12	11	ex 108	. . 3 ⊢ (𝜑 → (DECID 𝜓 → DECID (𝜑 → 𝜓)))
13		ax-in2 545	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
14	13	a1d 22	. . . 4 ⊢ (¬ 𝜑 → (DECID 𝜓 → (𝜑 → 𝜓)))
15		orc 633	. . . . 5 ⊢ ((𝜑 → 𝜓) → ((𝜑 → 𝜓) ∨ ¬ (𝜑 → 𝜓)))
16	15, 10	sylibr 137	. . . 4 ⊢ ((𝜑 → 𝜓) → DECID (𝜑 → 𝜓))
17	14, 16	syl6 29	. . 3 ⊢ (¬ 𝜑 → (DECID 𝜓 → DECID (𝜑 → 𝜓)))
18	12, 17	jaoi 636	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (DECID 𝜓 → DECID (𝜑 → 𝜓)))
19	1, 18	sylbi 114	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 629  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  pm4.79dc  809  pm5.11dc  815  dcbi  844  annimdc  845