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Theorem nfiunxy 3834
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 3810 . 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 2273 . . . 4  |-  F/ y  z  e.  B
52, 4nfrexxy 2470 . . 3  |-  F/ y E. x  e.  A  z  e.  B
65nfab 2284 . 2  |-  F/_ y { z  |  E. x  e.  A  z  e.  B }
71, 6nfcxfr 2276 1  |-  F/_ y U_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1480   {cab 2123   F/_wnfc 2266   E.wrex 2415   U_ciun 3808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-iun 3810
This theorem is referenced by:  iunab  3854
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