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Theorem ralun 3472
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 3471 . 2  |-  ( A. x  e.  ( A  u.  B ) ph  <->  ( A. x  e.  A  ph  /\  A. x  e.  B  ph ) )
21biimpri 198 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 359   A.wral 2649    u. cun 3261
This theorem is referenced by:  ac6sfi  7287  frfi  7288  fpwwe2lem13  8450  drsdirfi  14322  lbsextlem4  16160  fbun  17793  filcon  17836  cnmpt2pc  18824  chtub  20863  eupap1  21546  prsiga  24310  kelac1  26830
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-v 2901  df-un 3268
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