Theorem nfdc 1705

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 840	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		nfdc.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfn 1704	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
4	2, 3	nfor 1620	. 2 ⊢ Ⅎ𝑥(𝜑 ∨ ¬ 𝜑)
5	1, 4	nfxfr 1520	1 ⊢ Ⅎ𝑥DECID 𝜑

Syntax hints:  ¬ wn 3   ∨ wo 713  DECID wdc 839  Ⅎwnf 1506

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-gen 1495  ax-ie2 1540  ax-4 1556  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-dc 840  df-tru 1398  df-fal 1401  df-nf 1507

This theorem is referenced by:  19.32dc  1725  finexdc  7073  ssfirab  7109  opabfi  7111  dcfi  7159  exfzdc  10458  zsupcllemstep  10461  infssuzex  10465  nfsum1  11882  nfsum  11883  nfcprod1  12080  nfcprod  12081  nnwosdc  12575  ctiunctlemudc  13023  iswomninnlem  16477