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Theorem ralunb 3444
 Description: Restricted quantification over a union. (Contributed by Scott Fenton, 12-Apr-2011.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
ralunb (x (AB)φ ↔ (x A φ x B φ))

Proof of Theorem ralunb
StepHypRef Expression
1 elun 3220 . . . . . 6 (x (AB) ↔ (x A x B))
21imbi1i 315 . . . . 5 ((x (AB) → φ) ↔ ((x A x B) → φ))
3 jaob 758 . . . . 5 (((x A x B) → φ) ↔ ((x Aφ) (x Bφ)))
42, 3bitri 240 . . . 4 ((x (AB) → φ) ↔ ((x Aφ) (x Bφ)))
54albii 1566 . . 3 (x(x (AB) → φ) ↔ x((x Aφ) (x Bφ)))
6 19.26 1593 . . 3 (x((x Aφ) (x Bφ)) ↔ (x(x Aφ) x(x Bφ)))
75, 6bitri 240 . 2 (x(x (AB) → φ) ↔ (x(x Aφ) x(x Bφ)))
8 df-ral 2619 . 2 (x (AB)φx(x (AB) → φ))
9 df-ral 2619 . . 3 (x A φx(x Aφ))
10 df-ral 2619 . . 3 (x B φx(x Bφ))
119, 10anbi12i 678 . 2 ((x A φ x B φ) ↔ (x(x Aφ) x(x Bφ)))
127, 8, 113bitr4i 268 1 (x (AB)φ ↔ (x A φ x B φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  ∀wral 2614   ∪ cun 3207 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214 This theorem is referenced by:  ralun  3445  ralprg  3775  raltpg  3777  ralunsn  3879  ssofss  4076
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