Theorem dcn 745

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		notnot1 559	. . . 4 ⊢ (φ → ¬ ¬ φ)
2	1	orim2i 677	. . 3 ⊢ ((¬ φ ∨ φ) → (¬ φ ∨ ¬ ¬ φ))
3	2	orcoms 648	. 2 ⊢ ((φ ∨ ¬ φ) → (¬ φ ∨ ¬ ¬ φ))
4		df-dc 742	. 2 ⊢ (DECID φ ↔ (φ ∨ ¬ φ))
5		df-dc 742	. 2 ⊢ (DECID ¬ φ ↔ (¬ φ ∨ ¬ ¬ φ))
6	3, 4, 5	3imtr4i 190	1 ⊢ (DECID φ → DECID ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 628  DECID wdc 741

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 629

This theorem depends on definitions:  df-bi 110  df-dc 742

This theorem is referenced by:  pm5.18dc  776  pm4.67dc  780  pm2.54dc  789  imordc  795  pm4.54dc  804  stabtestimpdc  823  annimdc  844  pm4.55dc  845  pm3.12dc  864  pm3.13dc  865  dn1dc  866  xor3dc  1275  dfbi3dc  1285  dcned  2209