Theorem nfdc 1659

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 835	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		nfdc.1	. . 3 |- F/ x ph
3	2	nfn 1658	. . 3 |- F/ x -. ph
4	2, 3	nfor 1574	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1474	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 708  DECID wdc 834   F/wnf 1460

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-gen 1449  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-fal 1359  df-nf 1461

This theorem is referenced by:  19.32dc  1679  finexdc  6902  ssfirab  6933  dcfi  6980  exfzdc  10240  nfsum1  11364  nfsum  11365  nfcprod1  11562  nfcprod  11563  zsupcllemstep  11946  infssuzex  11950  nnwosdc  12040  ctiunctlemudc  12438  iswomninnlem  14800