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Theorem nfiun 3947
Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiun.1  |-  F/_ y A
nfiun.2  |-  F/_ y B
Assertion
Ref Expression
nfiun  |-  F/_ y U_ x  e.  A  B

Proof of Theorem nfiun
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-iun 3923 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
2 nfiun.1 . . . 4  |-  F/_ y A
3 nfiun.2 . . . . 5  |-  F/_ y B
43nfcri 2426 . . . 4  |-  F/ y  z  e.  B
52, 4nfrex 2611 . . 3  |-  F/ y E. x  e.  A  z  e.  B
65nfab 2436 . 2  |-  F/_ y { z  |  E. x  e.  A  z  e.  B }
71, 6nfcxfr 2429 1  |-  F/_ y U_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1696   {cab 2282   F/_wnfc 2419   E.wrex 2557   U_ciun 3921
This theorem is referenced by:  iunab  3964  disjxiun  4036  ovoliunnul  18882  iundisjfi  23378  iundisj2fi  23379  iundisjf  23380  iundisj2f  23381  trpredlem1  24301  trpredrec  24312  bnj1498  29407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-iun 3923
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