Theorem dcn 737

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 547	. . . 4 ⊢ (φ → ¬ ¬ φ)
2	1	orim2i 665	. . 3 ⊢ ((¬ φ ∨ φ) → (¬ φ ∨ ¬ ¬ φ))
3	2	orcoms 636	. 2 ⊢ ((φ ∨ ¬ φ) → (¬ φ ∨ ¬ ¬ φ))
4		df-dc 734	. 2 ⊢ (DECID φ ↔ (φ ∨ ¬ φ))
5		df-dc 734	. 2 ⊢ (DECID ¬ φ ↔ (¬ φ ∨ ¬ ¬ φ))
6	3, 4, 5	3imtr4i 190	1 ⊢ (DECID φ → DECID ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 616  DECID wdc 733

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 532  ax-in2 533  ax-io 617

This theorem depends on definitions:  df-bi 110  df-dc 734

This theorem is referenced by:  pm5.18dc  770  pm4.67dc  774  pm2.54dc  783  imordc  789  pm4.54dc  798  annimdc  833  pm4.55dc  834  pm3.12dc  853  pm3.13dc  854  dn1dc  855  xor3dc  1261  dfbi3dc  1271