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Theorem nfuni 3586
Description: Bound-variable hypothesis builder for union. (Contributed by NM, 30-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypothesis
Ref Expression
nfuni.1  |-  F/_ x A
Assertion
Ref Expression
nfuni  |-  F/_ x U. A

Proof of Theorem nfuni
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfuni2 3582 . 2  |-  U. A  =  { y  |  E. z  e.  A  y  e.  z }
2 nfuni.1 . . . 4  |-  F/_ x A
3 nfv 1421 . . . 4  |-  F/ x  y  e.  z
42, 3nfrexxy 2361 . . 3  |-  F/ x E. z  e.  A  y  e.  z
54nfab 2182 . 2  |-  F/_ x { y  |  E. z  e.  A  y  e.  z }
61, 5nfcxfr 2175 1  |-  F/_ x U. A
Colors of variables: wff set class
Syntax hints:   {cab 2026   F/_wnfc 2165   E.wrex 2307   U.cuni 3580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-uni 3581
This theorem is referenced by:  nfiota1  4869  nfrecs  5922
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