Theorem ralun 3363

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

Step	Hyp	Ref	Expression
1		ralunb 3362	. 2 |- ( A. x e. ( A u. B ) ph <-> ( A. x e. A ph /\ A. x e. B ph ) )
2	1	biimpri 133	1 |- ( ( A. x e. A ph /\ A. x e. B ph ) -> A. x e. ( A u. B ) ph )

Syntax hints:   -> wi 4   /\ wa 104   A.wral 2486   u. cun 3172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-un 3178

This theorem is referenced by:  omsinds  4688  ac6sfi  7021  fimax2gtrilemstep  7023  finomni  7268  uzsinds  10626  iseqf1olemqk  10689  seq3f1olemstep  10696  fimaxre2  11653  modfsummod  11884