Theorem nfdc 1549

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 743	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		nfdc.1	. . 3 |- F/ x ph
3	2	nfn 1548	. . 3 |- F/ x -. ph
4	2, 3	nfor 1466	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1363	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 629  DECID wdc 742   F/wnf 1349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-gen 1338  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-dc 743  df-tru 1246  df-fal 1249  df-nf 1350

This theorem is referenced by:  19.32dc  1569