Theorem ralun 3446

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 3445	. 2 ⊢ (∀x ∈ (A ∪ B)φ ↔ (∀x ∈ A φ ∧ ∀x ∈ B φ))
2	1	biimpri 197	1 ⊢ ((∀x ∈ A φ ∧ ∀x ∈ B φ) → ∀x ∈ (A ∪ B)φ)

Syntax hints:   → wi 4   ∧ wa 358  ∀wral 2615   ∪ cun 3208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215

This theorem is referenced by: (None)