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Theorem nfifd 3404
Description: Deduction version of nfif 3405. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
nfifd.2 (𝜑 → Ⅎ𝑥𝜓)
nfifd.3 (𝜑𝑥𝐴)
nfifd.4 (𝜑𝑥𝐵)
Assertion
Ref Expression
nfifd (𝜑𝑥if(𝜓, 𝐴, 𝐵))

Proof of Theorem nfifd
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-if 3380 . 2 if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓))}
2 nfv 1464 . . 3 𝑦𝜑
3 nfifd.3 . . . . . 6 (𝜑𝑥𝐴)
43nfcrd 2238 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐴)
5 nfifd.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
64, 5nfand 1503 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
7 nfifd.4 . . . . . 6 (𝜑𝑥𝐵)
87nfcrd 2238 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐵)
95nfnd 1590 . . . . 5 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
108, 9nfand 1503 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐵 ∧ ¬ 𝜓))
116, 10nford 1502 . . 3 (𝜑 → Ⅎ𝑥((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓)))
122, 11nfabd 2243 . 2 (𝜑𝑥{𝑦 ∣ ((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓))})
131, 12nfcxfrd 2223 1 (𝜑𝑥if(𝜓, 𝐴, 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 662  wnf 1392  wcel 1436  {cab 2071  wnfc 2212  ifcif 3379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-if 3380
This theorem is referenced by:  nfif  3405
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