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 843	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
2		nfdc.1	. . 3 |- F/ x ph
3	2	nfn 1706	. . 3 |- F/ x -. ph
4	2, 3	nfor 1623	. 2 |- F/ x ( ph \/ -. ph )
5	1, 4	nfxfr 1523	1 |- F/ xDECID ph

Syntax hints:   -. wn 3   \/ wo 716  DECID wdc 842   F/wnf 1509

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 717  ax-5 1496  ax-gen 1498  ax-ie2 1543  ax-4 1559  ax-ial 1583

This theorem depends on definitions:  df-bi 117  df-dc 843  df-tru 1401  df-fal 1404  df-nf 1510

This theorem is referenced by:  19.32dc  1727  finexdc  7160  ssfirab  7197  opabfi  7200  dcfi  7268  exfzdc  10586  zsupcllemstep  10589  infssuzex  10593  nfsum1  12041  nfsum  12042  nfcprod1  12240  nfcprod  12241  nnwosdc  12735  ctiunctlemudc  13188  iswomninnlem  16834