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Theorem nfiinya 3927
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 3901 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
2 nfiunya.1 . . . 4 𝑦𝐴
3 nfiunya.2 . . . . 5 𝑦𝐵
43nfcri 2323 . . . 4 𝑦 𝑧𝐵
52, 4nfralya 2527 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2334 . 2 𝑦{𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2326 1 𝑦 𝑥𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wcel 2158  {cab 2173  wnfc 2316  wral 2465   ciin 3899
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-iin 3901
This theorem is referenced by: (None)
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