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Theorem nfiun 5046
Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2380. See nfiung 5048 for a less restrictive version requiring more axioms. (Revised by GG, 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 5017 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
2 nfiun.1 . . . 4 𝑦𝐴
3 nfiun.2 . . . . 5 𝑦𝐵
43nfcri 2900 . . . 4 𝑦 𝑧𝐵
52, 4nfrexw 3319 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2914 . 2 𝑦{𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2906 1 𝑦 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  {cab 2717  wnfc 2893  wrex 3076   ciun 5015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-iun 5017
This theorem is referenced by:  iunab  5074  disjxiun  5163  ttrclselem1  9794  ttrclselem2  9795  ovoliunnul  25561  iundisjf  32611  iundisj2f  32612  iundisjfi  32801  iundisj2fi  32802  bnj1498  35037  ss2iundf  43621  nfcoll  44225  fnlimcnv  45588  fnlimfvre  45595  fnlimabslt  45600  smfaddlem1  46684  smflimlem6  46697  smflim  46698  smfmullem4  46715  smflim2  46727  smflimsup  46749  smfliminf  46752  fsupdm  46763  finfdm  46767
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