Theorem nfdc 1647

Description: If 𝑥 is not free in 𝜑, it is not free in DECID 𝜑. (Contributed by Jim Kingdon, 11-Mar-2018.)

Hypothesis

Ref	Expression
nfdc.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfdc	⊢ Ⅎ𝑥DECID 𝜑

Proof of Theorem nfdc

Step	Hyp	Ref	Expression
1		df-dc 825	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		nfdc.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfn 1646	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
4	2, 3	nfor 1562	. 2 ⊢ Ⅎ𝑥(𝜑 ∨ ¬ 𝜑)
5	1, 4	nfxfr 1462	1 ⊢ Ⅎ𝑥DECID 𝜑

Syntax hints:  ¬ wn 3   ∨ wo 698  DECID wdc 824  Ⅎwnf 1448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-gen 1437  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-fal 1349  df-nf 1449

This theorem is referenced by:  19.32dc  1667  finexdc  6868  ssfirab  6899  dcfi  6946  exfzdc  10175  nfsum1  11297  nfsum  11298  nfcprod1  11495  nfcprod  11496  zsupcllemstep  11878  infssuzex  11882  nnwosdc  11972  ctiunctlemudc  12370  iswomninnlem  13928