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Theorem nfiund 42186
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 4513 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
2 nfv 1841 . . 3 𝑧𝜑
3 nfiund.1 . . . 4 𝑥𝜑
4 nfiund.2 . . . 4 (𝜑𝑦𝐴)
5 nfiund.3 . . . . 5 (𝜑𝑦𝐵)
65nfcrd 2768 . . . 4 (𝜑 → Ⅎ𝑦 𝑧𝐵)
73, 4, 6nfrexd 3003 . . 3 (𝜑 → Ⅎ𝑦𝑥𝐴 𝑧𝐵)
82, 7nfabd 2782 . 2 (𝜑𝑦{𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
91, 8nfcxfrd 2761 1 (𝜑𝑦 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1706  wcel 1988  {cab 2606  wnfc 2749  wrex 2910   ciun 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-iun 4513
This theorem is referenced by: (None)
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