Theorem ralun 4151

Description: Restricted quantification over union. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
ralun	⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑)

Proof of Theorem ralun

Step	Hyp	Ref	Expression
1		ralunb 4150	. 2 ⊢ (∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑))
2	1	biimpri 228	1 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑)

Syntax hints:   → wi 4   ∧ wa 395  ∀wral 3044   ∪ cun 3903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-v 3440  df-un 3910

This theorem is referenced by:  f1ounsn  7213  ac6sfi  9189  frfi  9190  fpwwe2lem12  10555  modfsummod  15719  drsdirfi  18229  lbsextlem4  21086  fbun  23743  filconn  23786  cnmpopc  24838  chtub  27139  prsiga  34100  finixpnum  37587  poimirlem31  37633  poimirlem32  37634  kelac1  43039  cantnfresb  43300