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Theorem nfiinya 3812
Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiunya.1 𝑦𝐴
nfiunya.2 𝑦𝐵
Assertion
Ref Expression
nfiinya 𝑦 𝑥𝐴 𝐵
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfiinya
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-iin 3786 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
2 nfiunya.1 . . . 4 𝑦𝐴
3 nfiunya.2 . . . . 5 𝑦𝐵
43nfcri 2252 . . . 4 𝑦 𝑧𝐵
52, 4nfralya 2450 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2263 . 2 𝑦{𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2255 1 𝑦 𝑥𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wcel 1465  {cab 2103  wnfc 2245  wral 2393   ciin 3784
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-iin 3786
This theorem is referenced by: (None)
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