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Theorem nfiundg 46080
Description: Bound-variable hypothesis builder for indexed union. Usage of this theorem is discouraged because it depends on ax-13 2372, see nfiund 46079 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 4920 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
2 nfv 1922 . . 3 𝑧𝜑
3 nfiundg.1 . . . 4 𝑥𝜑
4 nfiundg.2 . . . 4 (𝜑𝑦𝐴)
5 nfiundg.3 . . . . 5 (𝜑𝑦𝐵)
65nfcrd 2894 . . . 4 (𝜑 → Ⅎ𝑦 𝑧𝐵)
73, 4, 6nfrexdg 3235 . . 3 (𝜑 → Ⅎ𝑦𝑥𝐴 𝑧𝐵)
82, 7nfabd 2930 . 2 (𝜑𝑦{𝑧 ∣ ∃𝑥𝐴 𝑧𝐵})
91, 8nfcxfrd 2904 1 (𝜑𝑦 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1791  wcel 2111  {cab 2715  wnfc 2885  wrex 3063   ciun 4918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-13 2372  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-iun 4920
This theorem is referenced by: (None)
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