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Theorem nfiun 5028
Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2372. See nfiung 5030 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)
Hypotheses
Ref Expression
nfiun.1 𝑦𝐴
nfiun.2 𝑦𝐵
Assertion
Ref Expression
nfiun 𝑦 𝑥𝐴 𝐵
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfiun
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-iun 5000 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
2 nfiun.1 . . . 4 𝑦𝐴
3 nfiun.2 . . . . 5 𝑦𝐵
43nfcri 2891 . . . 4 𝑦 𝑧𝐵
52, 4nfrexw 3311 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2910 . 2 𝑦{𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2902 1 𝑦 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  {cab 2710  wnfc 2884  wrex 3071   ciun 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-iun 5000
This theorem is referenced by:  iunab  5055  disjxiun  5146  ttrclselem1  9720  ttrclselem2  9721  ovoliunnul  25024  iundisjf  31820  iundisj2f  31821  iundisjfi  32007  iundisj2fi  32008  bnj1498  34072  ss2iundf  42410  nfcoll  43015  fnlimcnv  44383  fnlimfvre  44390  fnlimabslt  44395  smfaddlem1  45479  smflimlem6  45492  smflim  45493  smfmullem4  45510  smflim2  45522  smflimsup  45544  smfliminf  45547  fsupdm  45558  finfdm  45562
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