Theorem nfdc 1682

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 1681	. . 3 |- F/ x -. ph
4	2, 3	nfor 1597	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1497	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 710  DECID wdc 836   F/wnf 1483

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 1470  ax-gen 1472  ax-ie2 1517  ax-4 1533  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-fal 1379  df-nf 1484

This theorem is referenced by:  19.32dc  1702  finexdc  7001  ssfirab  7035  opabfi  7037  dcfi  7085  exfzdc  10371  zsupcllemstep  10374  infssuzex  10378  nfsum1  11700  nfsum  11701  nfcprod1  11898  nfcprod  11899  nnwosdc  12393  ctiunctlemudc  12841  iswomninnlem  16025