Theorem nfdc 1637

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 820	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		nfdc.1	. . 3 |- F/ x ph
3	2	nfn 1636	. . 3 |- F/ x -. ph
4	2, 3	nfor 1553	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1450	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 697  DECID wdc 819   F/wnf 1436

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-gen 1425  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-fal 1337  df-nf 1437

This theorem is referenced by:  19.32dc  1657  finexdc  6789  ssfirab  6815  exfzdc  10010  nfsum1  11118  nfsum  11119  nfcprod1  11316  nfcprod  11317  zsupcllemstep  11627  infssuzex  11631  ctiunctlemudc  11939