Theorem dcn 746

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 559	. . . 4 ⊢ (𝜑 → ¬ ¬ 𝜑)
2	1	orim2i 678	. . 3 ⊢ ((¬ 𝜑 ∨ 𝜑) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
3	2	orcoms 649	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (¬ 𝜑 ∨ ¬ ¬ 𝜑))
4		df-dc 743	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
5		df-dc 743	. 2 ⊢ (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
6	3, 4, 5	3imtr4i 190	1 ⊢ (DECID 𝜑 → DECID ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∨ 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:  pm5.18dc  777  pm4.67dc  781  pm2.54dc  790  imordc  796  pm4.54dc  805  stabtestimpdc  824  annimdc  845  pm4.55dc  846  pm3.12dc  865  pm3.13dc  866  dn1dc  867  xor3dc  1278  dfbi3dc  1288  dcned  2212