Theorem nfdc 1589

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 776	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		nfdc.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	nfn 1588	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
4	2, 3	nfor 1506	. 2 ⊢ Ⅎ𝑥(𝜑 ∨ ¬ 𝜑)
5	1, 4	nfxfr 1403	1 ⊢ Ⅎ𝑥DECID 𝜑

Syntax hints:  ¬ wn 3   ∨ wo 661  DECID wdc 775  Ⅎwnf 1389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-gen 1378  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-dc 776  df-tru 1287  df-fal 1290  df-nf 1390

This theorem is referenced by:  19.32dc  1609  exfzdc  9249  zsupcllemstep  10341  infssuzex  10345