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Theorem intun 3620
Description: The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)
Assertion
Ref Expression
intun (AB) = ( A B)

Proof of Theorem intun
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.26 1350 . . . 4 (y((y Ax y) (y Bx y)) ↔ (y(y Ax y) y(y Bx y)))
2 elun 3061 . . . . . . 7 (y (AB) ↔ (y A y B))
32imbi1i 227 . . . . . 6 ((y (AB) → x y) ↔ ((y A y B) → x y))
4 jaob 618 . . . . . 6 (((y A y B) → x y) ↔ ((y Ax y) (y Bx y)))
53, 4bitri 173 . . . . 5 ((y (AB) → x y) ↔ ((y Ax y) (y Bx y)))
65albii 1339 . . . 4 (y(y (AB) → x y) ↔ y((y Ax y) (y Bx y)))
7 vex 2538 . . . . . 6 x V
87elint 3595 . . . . 5 (x Ay(y Ax y))
97elint 3595 . . . . 5 (x By(y Bx y))
108, 9anbi12i 436 . . . 4 ((x A x B) ↔ (y(y Ax y) y(y Bx y)))
111, 6, 103bitr4i 201 . . 3 (y(y (AB) → x y) ↔ (x A x B))
127elint 3595 . . 3 (x (AB) ↔ y(y (AB) → x y))
13 elin 3103 . . 3 (x ( A B) ↔ (x A x B))
1411, 12, 133bitr4i 201 . 2 (x (AB) ↔ x ( A B))
1514eqriv 2019 1 (AB) = ( A B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 616  wal 1226   = wceq 1228   wcel 1374  cun 2892  cin 2893   cint 3589
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-int 3590
This theorem is referenced by:  intunsn  3627  riinint  4520
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