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Theorem nfiund 42219
 Description: Bound-variable hypothesis builder for indexed union. (Contributed by Emmett Weisz, 6-Dec-2019.)
Hypotheses
Ref Expression
nfiund.1 𝑥𝜑
nfiund.2 (𝜑𝑦𝐴)
nfiund.3 (𝜑𝑦𝐵)
Assertion
Ref Expression
nfiund (𝜑𝑦 𝑥𝐴 𝐵)

Proof of Theorem nfiund
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-iun 4457 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
2 nfv 1830 . . 3 𝑧𝜑
3 nfiund.1 . . . 4 𝑥𝜑
4 nfiund.2 . . . 4 (𝜑𝑦𝐴)
5 nfiund.3 . . . . 5 (𝜑𝑦𝐵)
65nfcrd 2757 . . . 4 (𝜑 → Ⅎ𝑦 𝑧𝐵)
73, 4, 6nfrexd 2989 . . 3 (𝜑 → Ⅎ𝑦𝑥𝐴 𝑧𝐵)
82, 7nfabd 2771 . 2 (𝜑𝑦{𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
91, 8nfcxfrd 2750 1 (𝜑𝑦 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  Ⅎwnf 1699   ∈ wcel 1977  {cab 2596  Ⅎwnfc 2738  ∃wrex 2897  ∪ ciun 4455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-iun 4457 This theorem is referenced by: (None)
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