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Theorem nfiun 4359
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 4333 . 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 2622 . . . 4  |-  F/ y  z  e.  B
52, 4nfrex 2930 . . 3  |-  F/ y E. x  e.  A  z  e.  B
65nfab 2633 . 2  |-  F/_ y { z  |  E. x  e.  A  z  e.  B }
71, 6nfcxfr 2627 1  |-  F/_ y U_ x  e.  A  B
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767   {cab 2452   F/_wnfc 2615   E.wrex 2818   U_ciun 4331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-iun 4333
This theorem is referenced by:  iunab  4377  disjxiun  4450  ovoliunnul  21786  iundisjf  27271  iundisj2f  27272  iundisjfi  27424  iundisj2fi  27425  trpredlem1  29237  trpredrec  29248  bnj1498  33597
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