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Theorem nfiunya 3848
Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiunya.1  |-  F/_ y A
nfiunya.2  |-  F/_ y B
Assertion
Ref Expression
nfiunya  |-  F/_ y U_ x  e.  A  B
Distinct variable group:    x, A
Allowed substitution hints:    A( y)    B( x, y)

Proof of Theorem nfiunya
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-iun 3822 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
2 nfiunya.1 . . . 4  |-  F/_ y A
3 nfiunya.2 . . . . 5  |-  F/_ y B
43nfcri 2276 . . . 4  |-  F/ y  z  e.  B
52, 4nfrexya 2477 . . 3  |-  F/ y E. x  e.  A  z  e.  B
65nfab 2287 . 2  |-  F/_ y { z  |  E. x  e.  A  z  e.  B }
71, 6nfcxfr 2279 1  |-  F/_ y U_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1481   {cab 2126   F/_wnfc 2269   E.wrex 2418   U_ciun 3820
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-iun 3822
This theorem is referenced by: (None)
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