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Theorem nfiund 42946
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 4657 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
2 nfv 1995 . . 3 𝑧𝜑
3 nfiund.1 . . . 4 𝑥𝜑
4 nfiund.2 . . . 4 (𝜑𝑦𝐴)
5 nfiund.3 . . . . 5 (𝜑𝑦𝐵)
65nfcrd 2920 . . . 4 (𝜑 → Ⅎ𝑦 𝑧𝐵)
73, 4, 6nfrexd 3154 . . 3 (𝜑 → Ⅎ𝑦𝑥𝐴 𝑧𝐵)
82, 7nfabd 2934 . 2 (𝜑𝑦{𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
91, 8nfcxfrd 2912 1 (𝜑𝑦 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1856  wcel 2145  {cab 2757  wnfc 2900  wrex 3062   ciun 4655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-iun 4657
This theorem is referenced by: (None)
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