Theorem ralun 4150

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 4149	. 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 3051   ∪ cun 3899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-v 3442  df-un 3906

This theorem is referenced by:  f1ounsn  7218  ac6sfi  9184  frfi  9185  fpwwe2lem12  10553  modfsummod  15717  drsdirfi  18228  lbsextlem4  21116  fbun  23784  filconn  23827  cnmpopc  24878  chtub  27179  prsiga  34288  finixpnum  37806  poimirlem31  37852  poimirlem32  37853  kelac1  43305  cantnfresb  43566