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Theorem nfun 3921
Description: Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfun.1 𝑥𝐴
nfun.2 𝑥𝐵
Assertion
Ref Expression
nfun 𝑥(𝐴𝐵)

Proof of Theorem nfun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-un 3729 . 2 (𝐴𝐵) = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
2 nfun.1 . . . . 5 𝑥𝐴
32nfcri 2907 . . . 4 𝑥 𝑦𝐴
4 nfun.2 . . . . 5 𝑥𝐵
54nfcri 2907 . . . 4 𝑥 𝑦𝐵
63, 5nfor 1986 . . 3 𝑥(𝑦𝐴𝑦𝐵)
76nfab 2918 . 2 𝑥{𝑦 ∣ (𝑦𝐴𝑦𝐵)}
81, 7nfcxfr 2911 1 𝑥(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wo 828  wcel 2145  {cab 2757  wnfc 2900  cun 3722
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 829  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-un 3729
This theorem is referenced by:  nfsymdif  3998  csbun  4154  iunxdif3  4741  nfsuc  5940  nfsup  8514  nfdju  8935  iunconn  21453  ordtconnlem1  30311  esumsplit  30456  measvuni  30618  bnj958  31349  bnj1000  31350  bnj1408  31443  bnj1446  31452  bnj1447  31453  bnj1448  31454  bnj1466  31460  bnj1467  31461  nosupbnd2  32200  poimirlem16  33759  poimirlem19  33762  pimrecltpos  41440
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