Theorem nfdc 1707

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 842	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		nfdc.1	. . 3 |- F/ x ph
3	2	nfn 1706	. . 3 |- F/ x -. ph
4	2, 3	nfor 1622	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1522	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 715  DECID wdc 841   F/wnf 1508

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-gen 1497  ax-ie2 1542  ax-4 1558  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-fal 1403  df-nf 1509

This theorem is referenced by:  19.32dc  1727  finexdc  7091  ssfirab  7128  opabfi  7131  dcfi  7179  exfzdc  10485  zsupcllemstep  10488  infssuzex  10492  nfsum1  11916  nfsum  11917  nfcprod1  12114  nfcprod  12115  nnwosdc  12609  ctiunctlemudc  13057  iswomninnlem  16653