Theorem nfdc 1639

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 821	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		nfdc.1	. . 3 |- F/ x ph
3	2	nfn 1638	. . 3 |- F/ x -. ph
4	2, 3	nfor 1554	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1454	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 698  DECID wdc 820   F/wnf 1440

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-gen 1429  ax-ie2 1474  ax-4 1490  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-fal 1341  df-nf 1441

This theorem is referenced by:  19.32dc  1659  finexdc  6840  ssfirab  6871  exfzdc  10121  nfsum1  11235  nfsum  11236  nfcprod1  11433  nfcprod  11434  zsupcllemstep  11813  infssuzex  11817  ctiunctlemudc  12138  iswomninnlem  13583