Theorem nfdc 1652

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 830	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		nfdc.1	. . 3 |- F/ x ph
3	2	nfn 1651	. . 3 |- F/ x -. ph
4	2, 3	nfor 1567	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1467	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 703  DECID wdc 829   F/wnf 1453

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-gen 1442  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-fal 1354  df-nf 1454

This theorem is referenced by:  19.32dc  1672  finexdc  6880  ssfirab  6911  dcfi  6958  exfzdc  10196  nfsum1  11319  nfsum  11320  nfcprod1  11517  nfcprod  11518  zsupcllemstep  11900  infssuzex  11904  nnwosdc  11994  ctiunctlemudc  12392  iswomninnlem  14081