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Theorem nfifd 4549
Description: Deduction form of nfif 4550. (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 4522 . 2 if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦𝐵𝜓) → (𝑦𝐴𝜓))}
2 nfv 1909 . . 3 𝑦𝜑
3 nfifd.4 . . . . . 6 (𝜑𝑥𝐵)
43nfcrd 2884 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐵)
5 nfifd.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
64, 5nfimd 1889 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐵𝜓))
7 nfifd.3 . . . . . 6 (𝜑𝑥𝐴)
87nfcrd 2884 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐴)
98, 5nfand 1892 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
106, 9nfimd 1889 . . 3 (𝜑 → Ⅎ𝑥((𝑦𝐵𝜓) → (𝑦𝐴𝜓)))
112, 10nfabdw 2918 . 2 (𝜑𝑥{𝑦 ∣ ((𝑦𝐵𝜓) → (𝑦𝐴𝜓))})
121, 11nfcxfrd 2894 1 (𝜑𝑥if(𝜓, 𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1777  wcel 2098  {cab 2701  wnfc 2875  ifcif 4520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-if 4521
This theorem is referenced by:  nfif  4550  nfxnegd  44636
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