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

Theorem ralun 3938
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 3937 . 2 (∀𝑥 ∈ (𝐴𝐵)𝜑 ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑))
21biimpri 218 1 ((∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 𝜑) → ∀𝑥 ∈ (𝐴𝐵)𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wral 3050  cun 3713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-v 3342  df-un 3720
This theorem is referenced by:  ac6sfi  8371  frfi  8372  fpwwe2lem13  9676  modfsummod  14745  drsdirfi  17159  lbsextlem4  19383  fbun  21865  filconn  21908  cnmpt2pc  22948  chtub  25157  prsiga  30524  finixpnum  33725  poimirlem31  33771  poimirlem32  33772  kelac1  38153
  Copyright terms: Public domain W3C validator