Theorem nfdc 1707

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 843	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		nfdc.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfn 1706	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
4	2, 3	nfor 1623	. 2 ⊢ Ⅎ𝑥(𝜑 ∨ ¬ 𝜑)
5	1, 4	nfxfr 1523	1 ⊢ Ⅎ𝑥DECID 𝜑

Syntax hints:  ¬ wn 3   ∨ wo 716  DECID wdc 842  Ⅎwnf 1509

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-gen 1498  ax-ie2 1543  ax-4 1559  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-dc 843  df-tru 1401  df-fal 1404  df-nf 1510

This theorem is referenced by:  19.32dc  1727  finexdc  7135  ssfirab  7172  opabfi  7175  dcfi  7223  exfzdc  10532  zsupcllemstep  10535  infssuzex  10539  nfsum1  11979  nfsum  11980  nfcprod1  12178  nfcprod  12179  nnwosdc  12673  ctiunctlemudc  13121  iswomninnlem  16765