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Theorem nfiunxy 3897
Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiunxy.1  |-  F/_ y A
nfiunxy.2  |-  F/_ y B
Assertion
Ref Expression
nfiunxy  |-  F/_ y U_ x  e.  A  B
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)

Proof of Theorem nfiunxy
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-iun 3873 . 2  |-  U_ x  e.  A  B  =  { z  |  E. x  e.  A  z  e.  B }
2 nfiunxy.1 . . . 4  |-  F/_ y A
3 nfiunxy.2 . . . . 5  |-  F/_ y B
43nfcri 2306 . . . 4  |-  F/ y  z  e.  B
52, 4nfrexxy 2509 . . 3  |-  F/ y E. x  e.  A  z  e.  B
65nfab 2317 . 2  |-  F/_ y { z  |  E. x  e.  A  z  e.  B }
71, 6nfcxfr 2309 1  |-  F/_ y U_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    e. wcel 2141   {cab 2156   F/_wnfc 2299   E.wrex 2449   U_ciun 3871
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-iun 3873
This theorem is referenced by:  iunab  3917
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