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Theorem nfiinxy 3675
 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 x A 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 3651 . 2 x A B = {zx A z B}
2 nfiunxy.1 . . . 4 yA
3 nfiunxy.2 . . . . 5 yB
43nfcri 2169 . . . 4 y z B
52, 4nfralxy 2354 . . 3 yx A z B
65nfab 2179 . 2 y{zx A z B}
71, 6nfcxfr 2172 1 y x A B
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1390  {cab 2023  Ⅎwnfc 2162  ∀wral 2300  ∩ ciin 3649 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-iin 3651 This theorem is referenced by:  iinab  3709
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