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Theorem nfdc 1669
Description: If 𝑥 is not free in 𝜑, it is not free in DECID 𝜑. (Contributed by Jim Kingdon, 11-Mar-2018.)
Hypothesis
Ref Expression
nfdc.1 𝑥𝜑
Assertion
Ref Expression
nfdc 𝑥DECID 𝜑

Proof of Theorem nfdc
StepHypRef Expression
1 df-dc 836 . 2 (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2 nfdc.1 . . 3 𝑥𝜑
32nfn 1668 . . 3 𝑥 ¬ 𝜑
42, 3nfor 1584 . 2 𝑥(𝜑 ∨ ¬ 𝜑)
51, 4nfxfr 1484 1 𝑥DECID 𝜑
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wo 709  DECID wdc 835  wnf 1470
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-gen 1459  ax-ie2 1504  ax-4 1520  ax-ial 1544
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1366  df-fal 1369  df-nf 1471
This theorem is referenced by:  19.32dc  1689  finexdc  6916  ssfirab  6947  dcfi  6994  exfzdc  10254  nfsum1  11378  nfsum  11379  nfcprod1  11576  nfcprod  11577  zsupcllemstep  11960  infssuzex  11964  nnwosdc  12054  ctiunctlemudc  12452  iswomninnlem  15094
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