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Theorem ralun 3183
Description: Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
ralun ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑) → ∀𝑥 ∈ (𝐴𝐵)𝜑)

Proof of Theorem ralun
StepHypRef Expression
1 ralunb 3182 . 2 (∀𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑))
21biimpri 132 1 ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑) → ∀𝑥 ∈ (𝐴𝐵)𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wral 2360  cun 2998
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2622  df-un 3004
This theorem is referenced by:  omsinds  4448  ac6sfi  6668  fimax2gtrilemstep  6670  finomni  6857  uzsinds  9909  iseqf1olemqk  9984  seq3f1olemstep  9991  fimaxre2  10719  modfsummod  10913
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