Theorem nfdc 1594

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 781	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		nfdc.1	. . 3 |- F/ x ph
3	2	nfn 1593	. . 3 |- F/ x -. ph
4	2, 3	nfor 1511	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1408	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 664  DECID wdc 780   F/wnf 1394

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-gen 1383  ax-ie2 1428  ax-4 1445  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-fal 1295  df-nf 1395

This theorem is referenced by:  19.32dc  1614  finexdc  6608  ssfirab  6633  exfzdc  9639  nfsum1  10732  nfsum  10733  zsupcllemstep  11206  infssuzex  11210