Theorem nfdc 1683

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 837	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		nfdc.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfn 1682	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
4	2, 3	nfor 1598	. 2 ⊢ Ⅎ𝑥(𝜑 ∨ ¬ 𝜑)
5	1, 4	nfxfr 1498	1 ⊢ Ⅎ𝑥DECID 𝜑

Syntax hints:  ¬ wn 3   ∨ wo 710  DECID wdc 836  Ⅎwnf 1484

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 711  ax-5 1471  ax-gen 1473  ax-ie2 1518  ax-4 1534  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-fal 1379  df-nf 1485

This theorem is referenced by:  19.32dc  1703  finexdc  7014  ssfirab  7048  opabfi  7050  dcfi  7098  exfzdc  10391  zsupcllemstep  10394  infssuzex  10398  nfsum1  11742  nfsum  11743  nfcprod1  11940  nfcprod  11941  nnwosdc  12435  ctiunctlemudc  12883  iswomninnlem  16129