Theorem nfdc 1669

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 836	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		nfdc.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfn 1668	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
4	2, 3	nfor 1584	. 2 ⊢ Ⅎ𝑥(𝜑 ∨ ¬ 𝜑)
5	1, 4	nfxfr 1484	1 ⊢ Ⅎ𝑥DECID 𝜑

Syntax hints:  ¬ wn 3   ∨ wo 709  DECID wdc 835  Ⅎwnf 1470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-gen 1459  ax-ie2 1504  ax-4 1520  ax-ial 1544

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1366  df-fal 1369  df-nf 1471

This theorem is referenced by:  19.32dc  1689  finexdc  6916  ssfirab  6947  dcfi  6994  exfzdc  10254  nfsum1  11378  nfsum  11379  nfcprod1  11576  nfcprod  11577  zsupcllemstep  11960  infssuzex  11964  nnwosdc  12054  ctiunctlemudc  12452  iswomninnlem  15094