Theorem nfdc 1638

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 821	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		nfdc.1	. . 3 |- F/ x ph
3	2	nfn 1637	. . 3 |- F/ x -. ph
4	2, 3	nfor 1554	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1451	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 698  DECID wdc 820   F/wnf 1437

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 1424  ax-gen 1426  ax-ie2 1471  ax-4 1488  ax-ial 1515

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-fal 1338  df-nf 1438

This theorem is referenced by:  19.32dc  1658  finexdc  6804  ssfirab  6830  exfzdc  10048  nfsum1  11157  nfsum  11158  nfcprod1  11355  nfcprod  11356  zsupcllemstep  11674  infssuzex  11678  ctiunctlemudc  11986  iswomninnlem  13417