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Theorem nfifd 3446
Description: Deduction version of nfif 3447. (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 3422 . 2 if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓))}
2 nfv 1476 . . 3 𝑦𝜑
3 nfifd.3 . . . . . 6 (𝜑𝑥𝐴)
43nfcrd 2254 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐴)
5 nfifd.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
64, 5nfand 1515 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
7 nfifd.4 . . . . . 6 (𝜑𝑥𝐵)
87nfcrd 2254 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐵)
95nfnd 1603 . . . . 5 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
108, 9nfand 1515 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐵 ∧ ¬ 𝜓))
116, 10nford 1514 . . 3 (𝜑 → Ⅎ𝑥((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓)))
122, 11nfabd 2259 . 2 (𝜑𝑥{𝑦 ∣ ((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓))})
131, 12nfcxfrd 2238 1 (𝜑𝑥if(𝜓, 𝐴, 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 670  wnf 1404  wcel 1448  {cab 2086  wnfc 2227  ifcif 3421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-if 3422
This theorem is referenced by:  nfif  3447
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