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Theorem ralun 3617
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 3616 . 2  |-  ( A. x  e.  ( A  u.  B ) ph  <->  ( A. x  e.  A  ph  /\  A. x  e.  B  ph ) )
21biimpri 206 1  |-  ( ( A. x  e.  A  ph 
/\  A. x  e.  B  ph )  ->  A. x  e.  ( A  u.  B
) ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wral 2746    u. cun 3404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ral 2751  df-v 3053  df-un 3411
This theorem is referenced by:  ac6sfi  7701  frfi  7702  fpwwe2lem13  8953  modfsummod  13633  drsdirfi  15707  lbsextlem4  17943  fbun  20449  filcon  20492  cnmpt2pc  21536  chtub  23627  eupap1  25122  prsiga  28315  finixpnum  30243  kelac1  31215
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