Theorem nfdc 1670

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 836	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		nfdc.1	. . 3 |- F/ x ph
3	2	nfn 1669	. . 3 |- F/ x -. ph
4	2, 3	nfor 1585	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1485	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 709  DECID wdc 835   F/wnf 1471

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 710  ax-5 1458  ax-gen 1460  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1472

This theorem is referenced by:  19.32dc  1690  finexdc  6958  ssfirab  6990  opabfi  6992  dcfi  7040  exfzdc  10307  nfsum1  11499  nfsum  11500  nfcprod1  11697  nfcprod  11698  zsupcllemstep  12082  infssuzex  12086  nnwosdc  12176  ctiunctlemudc  12594  iswomninnlem  15539