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Theorem nfifd 3576
Description: Deduction version of nfif 3577. (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 3550 . 2 if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓))}
2 nfv 1539 . . 3 𝑦𝜑
3 nfifd.3 . . . . . 6 (𝜑𝑥𝐴)
43nfcrd 2346 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐴)
5 nfifd.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
64, 5nfand 1579 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
7 nfifd.4 . . . . . 6 (𝜑𝑥𝐵)
87nfcrd 2346 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐵)
95nfnd 1668 . . . . 5 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
108, 9nfand 1579 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐵 ∧ ¬ 𝜓))
116, 10nford 1578 . . 3 (𝜑 → Ⅎ𝑥((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓)))
122, 11nfabd 2352 . 2 (𝜑𝑥{𝑦 ∣ ((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓))})
131, 12nfcxfrd 2330 1 (𝜑𝑥if(𝜓, 𝐴, 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wo 709  wnf 1471  wcel 2160  {cab 2175  wnfc 2319  ifcif 3549
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-if 3550
This theorem is referenced by:  nfif  3577
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