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Theorem nfiunya 3726
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 3700 . 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 2217 . . . 4  |-  F/ y  z  e.  B
52, 4nfrexya 2410 . . 3  |-  F/ y E. x  e.  A  z  e.  B
65nfab 2227 . 2  |-  F/_ y { z  |  E. x  e.  A  z  e.  B }
71, 6nfcxfr 2220 1  |-  F/_ y U_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1434   {cab 2069   F/_wnfc 2210   E.wrex 2354   U_ciun 3698
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-iun 3700
This theorem is referenced by: (None)
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