Theorem nfdc 1683

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 837	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		nfdc.1	. . 3 |- F/ x ph
3	2	nfn 1682	. . 3 |- F/ x -. ph
4	2, 3	nfor 1598	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1498	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 710  DECID wdc 836   F/wnf 1484

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-gen 1473  ax-ie2 1518  ax-4 1534  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-fal 1379  df-nf 1485

This theorem is referenced by:  19.32dc  1703  finexdc  7025  ssfirab  7059  opabfi  7061  dcfi  7109  exfzdc  10406  zsupcllemstep  10409  infssuzex  10413  nfsum1  11782  nfsum  11783  nfcprod1  11980  nfcprod  11981  nnwosdc  12475  ctiunctlemudc  12923  iswomninnlem  16190