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

Theorem nfiotadw 6498
Description: Deduction version of nfiotaw 6499. Version of nfiotad 6500 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 18-Feb-2013.) Avoid ax-13 2371. (Revised by Gino Giotto, 26-Jan-2024.)
Hypotheses
Ref Expression
nfiotadw.1 𝑦𝜑
nfiotadw.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfiotadw (𝜑𝑥(℩𝑦𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem nfiotadw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfiota2 6496 . 2 (℩𝑦𝜓) = {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)}
2 nfv 1917 . . . 4 𝑧𝜑
3 nfiotadw.1 . . . . 5 𝑦𝜑
4 nfiotadw.2 . . . . . 6 (𝜑 → Ⅎ𝑥𝜓)
5 nfvd 1918 . . . . . 6 (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
64, 5nfbid 1905 . . . . 5 (𝜑 → Ⅎ𝑥(𝜓𝑦 = 𝑧))
73, 6nfald 2321 . . . 4 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
82, 7nfabdw 2926 . . 3 (𝜑𝑥{𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
98nfunid 4914 . 2 (𝜑𝑥 {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
101, 9nfcxfrd 2902 1 (𝜑𝑥(℩𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539  wnf 1785  {cab 2709  wnfc 2883   cuni 4908  cio 6493
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-v 3476  df-in 3955  df-ss 3965  df-sn 4629  df-uni 4909  df-iota 6495
This theorem is referenced by:  nfiotaw  6499  nfriotadw  7375
  Copyright terms: Public domain W3C validator