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Theorem nfiund 45191
 Description: Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.) Add disjoint variable condition to avoid ax-13 2382. See nfiundg 45192 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Hypotheses
Ref Expression
nfiund.1 𝑥𝜑
nfiund.2 (𝜑𝑦𝐴)
nfiund.3 (𝜑𝑦𝐵)
Assertion
Ref Expression
nfiund (𝜑𝑦 𝑥𝐴 𝐵)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfiund
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-iun 4886 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
2 nfv 1915 . . 3 𝑧𝜑
3 nfiund.1 . . . 4 𝑥𝜑
4 nfiund.2 . . . 4 (𝜑𝑦𝐴)
5 nfiund.3 . . . . 5 (𝜑𝑦𝐵)
65nfcrd 2948 . . . 4 (𝜑 → Ⅎ𝑦 𝑧𝐵)
73, 4, 6nfrexd 3269 . . 3 (𝜑 → Ⅎ𝑦𝑥𝐴 𝑧𝐵)
82, 7nfabdw 2979 . 2 (𝜑𝑦{𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
91, 8nfcxfrd 2957 1 (𝜑𝑦 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Ⅎwnf 1785   ∈ wcel 2112  {cab 2779  Ⅎwnfc 2939  ∃wrex 3110  ∪ ciun 4884 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-iun 4886 This theorem is referenced by: (None)
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