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Theorem ralunb 3097
 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 3057 . . . . . 6 (x (AB) ↔ (x A x B))
21imbi1i 227 . . . . 5 ((x (AB) → φ) ↔ ((x A x B) → φ))
3 jaob 618 . . . . 5 (((x A x B) → φ) ↔ ((x Aφ) (x Bφ)))
42, 3bitri 173 . . . 4 ((x (AB) → φ) ↔ ((x Aφ) (x Bφ)))
54albii 1335 . . 3 (x(x (AB) → φ) ↔ x((x Aφ) (x Bφ)))
6 19.26 1346 . . 3 (x((x Aφ) (x Bφ)) ↔ (x(x Aφ) x(x Bφ)))
75, 6bitri 173 . 2 (x(x (AB) → φ) ↔ (x(x Aφ) x(x Bφ)))
8 df-ral 2285 . 2 (x (AB)φx(x (AB) → φ))
9 df-ral 2285 . . 3 (x A φx(x Aφ))
10 df-ral 2285 . . 3 (x B φx(x Bφ))
119, 10anbi12i 436 . 2 ((x A φ x B φ) ↔ (x(x Aφ) x(x Bφ)))
127, 8, 113bitr4i 201 1 (x (AB)φ ↔ (x A φ x B φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 616  ∀wal 1224   ∈ wcel 1370  ∀wral 2280   ∪ cun 2888 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-v 2533  df-un 2895 This theorem is referenced by:  ralun  3098  ralprg  3391  raltpg  3393  ralunsn  3538
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