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Theorem nfun 3752
 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 3564 . 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  Ⅎwnfc 2748   ∪ cun 3557 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 3564 This theorem is referenced by:  nfsymdif  3831  csbun  3986  iunxdif3  4577  nfsuc  5760  nfsup  8309  iunconn  21154  ordtconnlem1  29776  esumsplit  29920  measvuni  30082  bnj958  30753  bnj1000  30754  bnj1408  30847  bnj1446  30856  bnj1447  30857  bnj1448  30858  bnj1466  30864  bnj1467  30865  poimirlem16  33092  poimirlem19  33095  pimrecltpos  40252
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