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Theorem nfiundg 47975
Description: Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2365, see nfiund 47974 for a weaker version that does not require it. (Contributed by Emmett Weisz, 6-Dec-2019.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfiundg.1 𝑥𝜑
nfiundg.2 (𝜑𝑦𝐴)
nfiundg.3 (𝜑𝑦𝐵)
Assertion
Ref Expression
nfiundg (𝜑𝑦 𝑥𝐴 𝐵)

Proof of Theorem nfiundg
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-iun 4992 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
2 nfv 1909 . . 3 𝑧𝜑
3 nfiundg.1 . . . 4 𝑥𝜑
4 nfiundg.2 . . . 4 (𝜑𝑦𝐴)
5 nfiundg.3 . . . . 5 (𝜑𝑦𝐵)
65nfcrd 2886 . . . 4 (𝜑 → Ⅎ𝑦 𝑧𝐵)
73, 4, 6nfrexd 3363 . . 3 (𝜑 → Ⅎ𝑦𝑥𝐴 𝑧𝐵)
82, 7nfabd 2922 . 2 (𝜑𝑦{𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
91, 8nfcxfrd 2896 1 (𝜑𝑦 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1777  wcel 2098  {cab 2703  wnfc 2877  wrex 3064   ciun 4990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2365  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-iun 4992
This theorem is referenced by: (None)
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