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Theorem nfifd 4491
Description: Deduction form of nfif 4492. (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 dfif2 4465 . 2 if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦𝐵𝜓) → (𝑦𝐴𝜓))}
2 nfv 1906 . . 3 𝑦𝜑
3 nfifd.4 . . . . . 6 (𝜑𝑥𝐵)
43nfcrd 2966 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐵)
5 nfifd.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
64, 5nfimd 1886 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐵𝜓))
7 nfifd.3 . . . . . 6 (𝜑𝑥𝐴)
87nfcrd 2966 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐴)
98, 5nfand 1889 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
106, 9nfimd 1886 . . 3 (𝜑 → Ⅎ𝑥((𝑦𝐵𝜓) → (𝑦𝐴𝜓)))
112, 10nfabdw 2997 . 2 (𝜑𝑥{𝑦 ∣ ((𝑦𝐵𝜓) → (𝑦𝐴𝜓))})
121, 11nfcxfrd 2973 1 (𝜑𝑥if(𝜓, 𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wnf 1775  wcel 2105  {cab 2796  wnfc 2958  ifcif 4463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-if 4464
This theorem is referenced by:  nfif  4492  nfxnegd  41591
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