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Theorem nfiinxy 3913
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 3889 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
2 nfiunxy.1 . . . 4 𝑦𝐴
3 nfiunxy.2 . . . . 5 𝑦𝐵
43nfcri 2313 . . . 4 𝑦 𝑧𝐵
52, 4nfralxy 2515 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2324 . 2 𝑦{𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2316 1 𝑦 𝑥𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wcel 2148  {cab 2163  wnfc 2306  wral 2455   ciin 3887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-iin 3889
This theorem is referenced by:  iinab  3948
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