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Theorem nfiinxy 3712
Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiunxy.1  |-  F/_ y A
nfiunxy.2  |-  F/_ y B
Assertion
Ref Expression
nfiinxy  |-  F/_ y |^|_ x  e.  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 3688 . 2  |-  |^|_ x  e.  A  B  =  { z  |  A. x  e.  A  z  e.  B }
2 nfiunxy.1 . . . 4  |-  F/_ y A
3 nfiunxy.2 . . . . 5  |-  F/_ y B
43nfcri 2188 . . . 4  |-  F/ y  z  e.  B
52, 4nfralxy 2377 . . 3  |-  F/ y A. x  e.  A  z  e.  B
65nfab 2198 . 2  |-  F/_ y { z  |  A. x  e.  A  z  e.  B }
71, 6nfcxfr 2191 1  |-  F/_ y |^|_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1409   {cab 2042   F/_wnfc 2181   A.wral 2323   |^|_ciin 3686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-iin 3688
This theorem is referenced by:  iinab  3746
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