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Theorem nfun 3303
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 3145 . 2 (𝐴𝐵) = {𝑦 ∣ (𝑦𝐴𝑦𝐵)}
2 nfun.1 . . . . 5 𝑥𝐴
32nfcri 2323 . . . 4 𝑥 𝑦𝐴
4 nfun.2 . . . . 5 𝑥𝐵
54nfcri 2323 . . . 4 𝑥 𝑦𝐵
63, 5nfor 1584 . . 3 𝑥(𝑦𝐴𝑦𝐵)
76nfab 2334 . 2 𝑥{𝑦 ∣ (𝑦𝐴𝑦𝐵)}
81, 7nfcxfr 2326 1 𝑥(𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wo 709  wcel 2158  {cab 2173  wnfc 2316  cun 3139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-un 3145
This theorem is referenced by:  nfsuc  4420  nfdju  7055
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