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Theorem nfiunya 3877
 Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiunya.1 𝑦𝐴
nfiunya.2 𝑦𝐵
Assertion
Ref Expression
nfiunya 𝑦 𝑥𝐴 𝐵
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfiunya
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-iun 3851 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
2 nfiunya.1 . . . 4 𝑦𝐴
3 nfiunya.2 . . . . 5 𝑦𝐵
43nfcri 2293 . . . 4 𝑦 𝑧𝐵
52, 4nfrexya 2498 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2304 . 2 𝑦{𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2296 1 𝑦 𝑥𝐴 𝐵
 Colors of variables: wff set class Syntax hints:   ∈ wcel 2128  {cab 2143  Ⅎwnfc 2286  ∃wrex 2436  ∪ ciun 3849 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-iun 3851 This theorem is referenced by: (None)
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