Theorem nfdc 1711

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 847	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		nfdc.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfn 1710	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
4	2, 3	nfor 1627	. 2 ⊢ Ⅎ𝑥(𝜑 ∨ ¬ 𝜑)
5	1, 4	nfxfr 1527	1 ⊢ Ⅎ𝑥DECID 𝜑

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 720  DECID wdc 846  Ⅎwnf 1513

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-gen 1502  ax-ie2 1547  ax-4 1563  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-fal 1408  df-nf 1514

This theorem is used by:  19.32dc  1731  finexdc  7207  ssfirab  7244  opabfi  7247  dcfi  7315  exfzdc  10669  zsupcllemstep  10672  infssuzex  10676  nfsum1  12138  nfsum  12139  nfcprod1  12337  nfcprod  12338  nnwosdc  12832  ctiunctlemudc  13377  iswomninnlem  17197