Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralun Structured version   Visualization version   GIF version

Theorem ralun 3757
 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 3756 . 2 (∀𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑))
21biimpri 217 1 ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑) → ∀𝑥 ∈ (𝐴𝐵)𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wral 2896   ∪ cun 3538 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-un 3545 This theorem is referenced by:  ac6sfi  8089  frfi  8090  fpwwe2lem13  9343  modfsummod  14367  drsdirfi  16761  lbsextlem4  18982  fbun  21454  filcon  21497  cnmpt2pc  22535  chtub  24737  eupap1  26503  prsiga  29521  finixpnum  32564  poimirlem31  32610  poimirlem32  32611  kelac1  36651
 Copyright terms: Public domain W3C validator