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Theorem nfiunxy 3712
 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 3688 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
2 nfiunxy.1 . . . 4 𝑦𝐴
3 nfiunxy.2 . . . . 5 𝑦𝐵
43nfcri 2214 . . . 4 𝑦 𝑧𝐵
52, 4nfrexxy 2404 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2224 . 2 𝑦{𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2217 1 𝑦 𝑥𝐴 𝐵
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1434  {cab 2068  Ⅎwnfc 2207  ∃wrex 2350  ∪ ciun 3686 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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-iun 3688 This theorem is referenced by:  iunab  3732
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