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Theorem nfifd 4555
Description: Deduction form of nfif 4556. (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 4527 . 2 if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦𝐵𝜓) → (𝑦𝐴𝜓))}
2 nfv 1914 . . 3 𝑦𝜑
3 nfifd.4 . . . . . 6 (𝜑𝑥𝐵)
43nfcrd 2899 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐵)
5 nfifd.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
64, 5nfimd 1894 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐵𝜓))
7 nfifd.3 . . . . . 6 (𝜑𝑥𝐴)
87nfcrd 2899 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐴)
98, 5nfand 1897 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
106, 9nfimd 1894 . . 3 (𝜑 → Ⅎ𝑥((𝑦𝐵𝜓) → (𝑦𝐴𝜓)))
112, 10nfabdw 2927 . 2 (𝜑𝑥{𝑦 ∣ ((𝑦𝐵𝜓) → (𝑦𝐴𝜓))})
121, 11nfcxfrd 2904 1 (𝜑𝑥if(𝜓, 𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1783  wcel 2108  {cab 2714  wnfc 2890  ifcif 4525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-if 4526
This theorem is referenced by:  nfif  4556  nfxnegd  45452
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