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Theorem nfiunxy 3683
Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiunxy.1  |-  F/_ y A
nfiunxy.2  |-  F/_ y B
Assertion
Ref Expression
nfiunxy  |-  F/_ y U_ x  e.  A  B
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)

Proof of Theorem nfiunxy
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-iun 3659 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
2 nfiunxy.1 . . . 4  |-  F/_ y A
3 nfiunxy.2 . . . . 5  |-  F/_ y B
43nfcri 2172 . . . 4  |-  F/ y  z  e.  B
52, 4nfrexxy 2361 . . 3  |-  F/ y E. x  e.  A  z  e.  B
65nfab 2182 . 2  |-  F/_ y { z  |  E. x  e.  A  z  e.  B }
71, 6nfcxfr 2175 1  |-  F/_ y U_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1393   {cab 2026   F/_wnfc 2165   E.wrex 2307   U_ciun 3657
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-rex 2312  df-iun 3659
This theorem is referenced by:  iunab  3703
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