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Theorem nfifd 3382
Description: Deduction version of nfif 3383. (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 3359 . 2 if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓))}
2 nfv 1437 . . 3 𝑦𝜑
3 nfifd.3 . . . . . 6 (𝜑𝑥𝐴)
43nfcrd 2207 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐴)
5 nfifd.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
64, 5nfand 1476 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
7 nfifd.4 . . . . . 6 (𝜑𝑥𝐵)
87nfcrd 2207 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐵)
95nfnd 1563 . . . . 5 (𝜑 → Ⅎ𝑥 ¬ 𝜓)
108, 9nfand 1476 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐵 ∧ ¬ 𝜓))
116, 10nford 1475 . . 3 (𝜑 → Ⅎ𝑥((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓)))
122, 11nfabd 2212 . 2 (𝜑𝑥{𝑦 ∣ ((𝑦𝐴𝜓) ∨ (𝑦𝐵 ∧ ¬ 𝜓))})
131, 12nfcxfrd 2192 1 (𝜑𝑥if(𝜓, 𝐴, 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wo 639  wnf 1365  wcel 1409  {cab 2042  wnfc 2181  ifcif 3358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-if 3359
This theorem is referenced by:  nfif  3383
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