MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfifd Structured version   Visualization version   GIF version

Theorem nfifd 4477
Description: Deduction form of nfif 4478. (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 4451 . 2 if(𝜓, 𝐴, 𝐵) = {𝑦 ∣ ((𝑦𝐵𝜓) → (𝑦𝐴𝜓))}
2 nfv 1916 . . 3 𝑦𝜑
3 nfifd.4 . . . . . 6 (𝜑𝑥𝐵)
43nfcrd 2971 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐵)
5 nfifd.2 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
64, 5nfimd 1896 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐵𝜓))
7 nfifd.3 . . . . . 6 (𝜑𝑥𝐴)
87nfcrd 2971 . . . . 5 (𝜑 → Ⅎ𝑥 𝑦𝐴)
98, 5nfand 1899 . . . 4 (𝜑 → Ⅎ𝑥(𝑦𝐴𝜓))
106, 9nfimd 1896 . . 3 (𝜑 → Ⅎ𝑥((𝑦𝐵𝜓) → (𝑦𝐴𝜓)))
112, 10nfabdw 3003 . 2 (𝜑𝑥{𝑦 ∣ ((𝑦𝐵𝜓) → (𝑦𝐴𝜓))})
121, 11nfcxfrd 2981 1 (𝜑𝑥if(𝜓, 𝐴, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wnf 1785  wcel 2115  {cab 2802  wnfc 2962  ifcif 4449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-if 4450
This theorem is referenced by:  nfif  4478  nfxnegd  41941
  Copyright terms: Public domain W3C validator