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Theorem nfiunxy 3751
Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiunxy.1 𝑦𝐴
nfiunxy.2 𝑦𝐵
Assertion
Ref Expression
nfiunxy 𝑦 𝑥𝐴 𝐵
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfiunxy
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-iun 3727 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
2 nfiunxy.1 . . . 4 𝑦𝐴
3 nfiunxy.2 . . . . 5 𝑦𝐵
43nfcri 2222 . . . 4 𝑦 𝑧𝐵
52, 4nfrexxy 2415 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2233 . 2 𝑦{𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2225 1 𝑦 𝑥𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wcel 1438  {cab 2074  wnfc 2215  wrex 2360   ciun 3725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-iun 3727
This theorem is referenced by:  iunab  3771
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