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Theorem nfifd 4519
Description: Deduction form of nfif 4520. (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 4492 . 2 if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦𝐵𝜓) → (𝑦𝐴𝜓))}
2 nfv 1918 . . 3 𝑦𝜑
3 nfifd.4 . . . . . 6 (𝜑𝑥𝐵)
43nfcrd 2893 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐵)
5 nfifd.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
64, 5nfimd 1898 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐵𝜓))
7 nfifd.3 . . . . . 6 (𝜑𝑥𝐴)
87nfcrd 2893 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐴)
98, 5nfand 1901 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
106, 9nfimd 1898 . . 3 (𝜑 → Ⅎ𝑥((𝑦𝐵𝜓) → (𝑦𝐴𝜓)))
112, 10nfabdw 2927 . 2 (𝜑𝑥{𝑦 ∣ ((𝑦𝐵𝜓) → (𝑦𝐴𝜓))})
121, 11nfcxfrd 2903 1 (𝜑𝑥if(𝜓, 𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wnf 1786  wcel 2107  {cab 2710  wnfc 2884  ifcif 4490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-if 4491
This theorem is referenced by:  nfif  4520  nfxnegd  43766
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