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Theorem rexun 3117
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 2306 . 2 (x (AB)φx(x (AB) φ))
2 19.43 1516 . . 3 (x((x A φ) (x B φ)) ↔ (x(x A φ) x(x B φ)))
3 elun 3078 . . . . . 6 (x (AB) ↔ (x A x B))
43anbi1i 431 . . . . 5 ((x (AB) φ) ↔ ((x A x B) φ))
5 andir 731 . . . . 5 (((x A x B) φ) ↔ ((x A φ) (x B φ)))
64, 5bitri 173 . . . 4 ((x (AB) φ) ↔ ((x A φ) (x B φ)))
76exbii 1493 . . 3 (x(x (AB) φ) ↔ x((x A φ) (x B φ)))
8 df-rex 2306 . . . 4 (x A φx(x A φ))
9 df-rex 2306 . . . 4 (x B φx(x B φ))
108, 9orbi12i 680 . . 3 ((x A φ x B φ) ↔ (x(x A φ) x(x B φ)))
112, 7, 103bitr4i 201 . 2 (x(x (AB) φ) ↔ (x A φ x B φ))
121, 11bitri 173 1 (x (AB)φ ↔ (x A φ x B φ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   wo 628  wex 1378   wcel 1390  wrex 2301  cun 2909
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916
This theorem is referenced by:  rexprg  3413  rextpg  3415  iunxun  3726
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