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Theorem nfiunya 3756
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 3730 . 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 2222 . . . 4  |-  F/ y  z  e.  B
52, 4nfrexya 2417 . . 3  |-  F/ y E. x  e.  A  z  e.  B
65nfab 2233 . 2  |-  F/_ y { z  |  E. x  e.  A  z  e.  B }
71, 6nfcxfr 2225 1  |-  F/_ y U_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1438   {cab 2074   F/_wnfc 2215   E.wrex 2360   U_ciun 3728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-iun 3730
This theorem is referenced by: (None)
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