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Theorem ralun 3228
Description: Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
ralun  |-  ( ( A. x  e.  A  ph 
/\  A. x  e.  B  ph )  ->  A. x  e.  ( A  u.  B
) ph )

Proof of Theorem ralun
StepHypRef Expression
1 ralunb 3227 . 2  |-  ( A. x  e.  ( A  u.  B ) ph  <->  ( A. x  e.  A  ph  /\  A. x  e.  B  ph ) )
21biimpri 132 1  |-  ( ( A. x  e.  A  ph 
/\  A. x  e.  B  ph )  ->  A. x  e.  ( A  u.  B
) ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wral 2393    u. cun 3039
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045
This theorem is referenced by:  omsinds  4505  ac6sfi  6760  fimax2gtrilemstep  6762  finomni  6980  uzsinds  10183  iseqf1olemqk  10235  seq3f1olemstep  10242  fimaxre2  10966  modfsummod  11195
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