Theorem ralun 4193

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 4192	. 2 ⊢ (∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑))
2	1	biimpri 227	1 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 𝜑) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)𝜑)

Syntax hints:   → wi 4   ∧ wa 397  ∀wral 3062   ∪ cun 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-un 3954

This theorem is referenced by:  ac6sfi  9287  frfi  9288  fpwwe2lem12  10637  modfsummod  15740  drsdirfi  18258  lbsextlem4  20774  fbun  23344  filconn  23387  cnmpopc  24444  chtub  26715  prsiga  33129  finixpnum  36473  poimirlem31  36519  poimirlem32  36520  kelac1  41805  cantnfresb  42074