Theorem dcn 779

Description: A decidable proposition is decidable when negated. (Contributed by Jim Kingdon, 25-Mar-2018.)

Assertion

Ref	Expression
dcn	⊢ (DECID 𝜑 → DECID ¬ 𝜑)

Proof of Theorem dcn

Step	Hyp	Ref	Expression
1		notnot 591	. . . 4 ⊢ (𝜑 → ¬ ¬ 𝜑)
2	1	orim2i 710	. . 3 ⊢ ((¬ 𝜑 ∨ 𝜑) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
3	2	orcoms 681	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4		df-dc 776	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5		df-dc 776	. 2 ⊢ (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
6	3, 4, 5	3imtr4i 199	1 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 661  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:  pm5.18dc  810  pm4.67dc  814  pm2.54dc  823  imordc  829  pm4.54dc  838  stabtestimpdc  857  annimdc  878  pm4.55dc  879  orandc  880  pm3.12dc  899  pm3.13dc  900  dn1dc  901  xor3dc  1318  dfbi3dc  1328  dcned  2251  gcdaddm  10375