Theorem nfdc 1705

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 840	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		nfdc.1	. . 3 |- F/ x ph
3	2	nfn 1704	. . 3 |- F/ x -. ph
4	2, 3	nfor 1620	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1520	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 713  DECID wdc 839   F/wnf 1506

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-gen 1495  ax-ie2 1540  ax-4 1556  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-dc 840  df-tru 1398  df-fal 1401  df-nf 1507

This theorem is referenced by:  19.32dc  1725  finexdc  7064  ssfirab  7098  opabfi  7100  dcfi  7148  exfzdc  10446  zsupcllemstep  10449  infssuzex  10453  nfsum1  11867  nfsum  11868  nfcprod1  12065  nfcprod  12066  nnwosdc  12560  ctiunctlemudc  13008  iswomninnlem  16417