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Theorem nfun 3753
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 3565 . 2 (𝐴𝐵) = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
2 nfun.1 . . . . 5 𝑥𝐴
32nfcri 2755 . . . 4 𝑥 𝑦𝐴
4 nfun.2 . . . . 5 𝑥𝐵
54nfcri 2755 . . . 4 𝑥 𝑦𝐵
63, 5nfor 1831 . . 3 𝑥(𝑦𝐴𝑦𝐵)
76nfab 2765 . 2 𝑥{𝑦 ∣ (𝑦𝐴𝑦𝐵)}
81, 7nfcxfr 2759 1 𝑥(𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wo 383  wcel 1987  {cab 2607  wnfc 2748  cun 3558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-un 3565
This theorem is referenced by:  nfsymdif  3832  csbun  3987  iunxdif3  4579  nfsuc  5765  nfsup  8317  iunconn  21171  ordtconnlem1  29794  esumsplit  29938  measvuni  30100  bnj958  30771  bnj1000  30772  bnj1408  30865  bnj1446  30874  bnj1447  30875  bnj1448  30876  bnj1466  30882  bnj1467  30883  poimirlem16  33096  poimirlem19  33099  pimrecltpos  40256
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