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Theorem nfiunxy 3845
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 3821 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
2 nfiunxy.1 . . . 4 𝑦𝐴
3 nfiunxy.2 . . . . 5 𝑦𝐵
43nfcri 2276 . . . 4 𝑦 𝑧𝐵
52, 4nfrexxy 2475 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2287 . 2 𝑦{𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2279 1 𝑦 𝑥𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wcel 1481  {cab 2126  wnfc 2269  wrex 2418   ciun 3819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-iun 3821
This theorem is referenced by:  iunab  3865
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