Theorem nfdc 1711

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 847	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		nfdc.1	. . 3 |- F/ x ph
3	2	nfn 1710	. . 3 |- F/ x -. ph
4	2, 3	nfor 1627	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1527	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 720  DECID wdc 846   F/wnf 1513

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-gen 1502  ax-ie2 1547  ax-4 1563  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-dc 847  df-tru 1405  df-fal 1408  df-nf 1514

This theorem is referenced by:  19.32dc  1731  finexdc  7197  ssfirab  7234  opabfi  7237  dcfi  7305  exfzdc  10637  zsupcllemstep  10640  infssuzex  10644  nfsum1  12100  nfsum  12101  nfcprod1  12299  nfcprod  12300  nnwosdc  12794  ctiunctlemudc  13306  iswomninnlem  17004