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Theorem rexun 3443
 Description: Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.)
Assertion
Ref Expression
rexun (x (AB)φ ↔ (x A φ x B φ))

Proof of Theorem rexun
StepHypRef Expression
1 df-rex 2620 . 2 (x (AB)φx(x (AB) φ))
2 19.43 1605 . . 3 (x((x A φ) (x B φ)) ↔ (x(x A φ) x(x B φ)))
3 elun 3220 . . . . . 6 (x (AB) ↔ (x A x B))
43anbi1i 676 . . . . 5 ((x (AB) φ) ↔ ((x A x B) φ))
5 andir 838 . . . . 5 (((x A x B) φ) ↔ ((x A φ) (x B φ)))
64, 5bitri 240 . . . 4 ((x (AB) φ) ↔ ((x A φ) (x B φ)))
76exbii 1582 . . 3 (x(x (AB) φ) ↔ x((x A φ) (x B φ)))
8 df-rex 2620 . . . 4 (x A φx(x A φ))
9 df-rex 2620 . . . 4 (x B φx(x B φ))
108, 9orbi12i 507 . . 3 ((x A φ x B φ) ↔ (x(x A φ) x(x B φ)))
112, 7, 103bitr4i 268 . 2 (x(x (AB) φ) ↔ (x A φ x B φ))
121, 11bitri 240 1 (x (AB)φ ↔ (x A φ x B φ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃wrex 2615   ∪ 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-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214 This theorem is referenced by:  rexprg  3776  rextpg  3778  iunxun  4047  pw1un  4163  nnadjoin  4520  tfinnn  4534  phiun  4614
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