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Theorem nfifd 3503
Description: Deduction version of nfif 3504. (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 3479 . 2 if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓))}
2 nfv 1509 . . 3 𝑦𝜑
3 nfifd.3 . . . . . 6 (𝜑𝑥𝐴)
43nfcrd 2296 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐴)
5 nfifd.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
64, 5nfand 1548 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
7 nfifd.4 . . . . . 6 (𝜑𝑥𝐵)
87nfcrd 2296 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐵)
95nfnd 1636 . . . . 5 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
108, 9nfand 1548 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐵 ∧ ¬ 𝜓))
116, 10nford 1547 . . 3 (𝜑 → Ⅎ𝑥((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓)))
122, 11nfabd 2301 . 2 (𝜑𝑥{𝑦 ∣ ((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓))})
131, 12nfcxfrd 2280 1 (𝜑𝑥if(𝜓, 𝐴, 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 698  wnf 1437  wcel 1481  {cab 2126  wnfc 2269  ifcif 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-if 3479
This theorem is referenced by:  nfif  3504
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