Theorem nfdc 1659

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 835	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		nfdc.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfn 1658	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
4	2, 3	nfor 1574	. 2 ⊢ Ⅎ𝑥(𝜑 ∨ ¬ 𝜑)
5	1, 4	nfxfr 1474	1 ⊢ Ⅎ𝑥DECID 𝜑

Syntax hints:  ¬ wn 3   ∨ wo 708  DECID wdc 834  Ⅎwnf 1460

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-gen 1449  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-fal 1359  df-nf 1461

This theorem is referenced by:  19.32dc  1679  finexdc  6904  ssfirab  6935  dcfi  6982  exfzdc  10242  nfsum1  11366  nfsum  11367  nfcprod1  11564  nfcprod  11565  zsupcllemstep  11948  infssuzex  11952  nnwosdc  12042  ctiunctlemudc  12440  iswomninnlem  14882