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 Description: Deduction version of nfiotaw 6302. Version of nfiotad 6303 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 18-Feb-2013.) (Revised by Gino Giotto, 26-Jan-2024.)
Hypotheses
Ref Expression
Assertion
Ref Expression
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfiota2 6299 . 2 (℩𝑦𝜓) = {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)}
2 nfv 1915 . . . 4 𝑧𝜑
3 nfiotadw.1 . . . . 5 𝑦𝜑
4 nfiotadw.2 . . . . . 6 (𝜑 → Ⅎ𝑥𝜓)
5 nfvd 1916 . . . . . 6 (𝜑 → Ⅎ𝑥 𝑦 = 𝑧)
64, 5nfbid 1903 . . . . 5 (𝜑 → Ⅎ𝑥(𝜓𝑦 = 𝑧))
73, 6nfald 2336 . . . 4 (𝜑 → Ⅎ𝑥𝑦(𝜓𝑦 = 𝑧))
82, 7nfabdw 2940 . . 3 (𝜑𝑥{𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
98nfunid 4807 . 2 (𝜑𝑥 {𝑧 ∣ ∀𝑦(𝜓𝑦 = 𝑧)})
101, 9nfcxfrd 2918 1 (𝜑𝑥(℩𝑦𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  Ⅎwnf 1785  {cab 2735  Ⅎwnfc 2899  ∪ cuni 4801  ℩cio 6296 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-in 3867  df-ss 3877  df-sn 4526  df-uni 4802  df-iota 6298 This theorem is referenced by:  nfiotaw  6302  nfriotadw  7121
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