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Theorem nfiinxy 3684
Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiunxy.1 𝑦𝐴
nfiunxy.2 𝑦𝐵
Assertion
Ref Expression
nfiinxy 𝑦 𝑥𝐴 𝐵
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfiinxy
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-iin 3660 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
2 nfiunxy.1 . . . 4 𝑦𝐴
3 nfiunxy.2 . . . . 5 𝑦𝐵
43nfcri 2172 . . . 4 𝑦 𝑧𝐵
52, 4nfralxy 2360 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2182 . 2 𝑦{𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2175 1 𝑦 𝑥𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wcel 1393  {cab 2026  wnfc 2165  wral 2306   ciin 3658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-iin 3660
This theorem is referenced by:  iinab  3718
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