Theorem nfdc 1647

Description: If x is not free in ph, it is not free in DECID ph. (Contributed by Jim Kingdon, 11-Mar-2018.)

Hypothesis

Ref	Expression
nfdc.1	|- F/ x ph

Assertion

Ref	Expression
nfdc	|- F/ xDECID ph

Proof of Theorem nfdc

Step	Hyp	Ref	Expression
1		df-dc 825	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		nfdc.1	. . 3 |- F/ x ph
3	2	nfn 1646	. . 3 |- F/ x -. ph
4	2, 3	nfor 1562	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1462	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 698  DECID wdc 824   F/wnf 1448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-gen 1437  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1346  df-fal 1349  df-nf 1449

This theorem is referenced by:  19.32dc  1667  finexdc  6868  ssfirab  6899  dcfi  6946  exfzdc  10175  nfsum1  11297  nfsum  11298  nfcprod1  11495  nfcprod  11496  zsupcllemstep  11878  infssuzex  11882  nnwosdc  11972  ctiunctlemudc  12370  iswomninnlem  13928