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Theorem intun 3958
 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) = (AB)

Proof of Theorem intun
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.26 1593 . . . 4 (y((y Ax y) (y Bx y)) ↔ (y(y Ax y) y(y Bx y)))
2 elun 3220 . . . . . . 7 (y (AB) ↔ (y A y B))
32imbi1i 315 . . . . . 6 ((y (AB) → x y) ↔ ((y A y B) → x y))
4 jaob 758 . . . . . 6 (((y A y B) → x y) ↔ ((y Ax y) (y Bx y)))
53, 4bitri 240 . . . . 5 ((y (AB) → x y) ↔ ((y Ax y) (y Bx y)))
65albii 1566 . . . 4 (y(y (AB) → x y) ↔ y((y Ax y) (y Bx y)))
7 vex 2862 . . . . . 6 x V
87elint 3932 . . . . 5 (x Ay(y Ax y))
97elint 3932 . . . . 5 (x By(y Bx y))
108, 9anbi12i 678 . . . 4 ((x A x B) ↔ (y(y Ax y) y(y Bx y)))
111, 6, 103bitr4i 268 . . 3 (y(y (AB) → x y) ↔ (x A x B))
127elint 3932 . . 3 (x (AB) ↔ y(y (AB) → x y))
13 elin 3219 . . 3 (x (AB) ↔ (x A x B))
1411, 12, 133bitr4i 268 . 2 (x (AB) ↔ x (AB))
1514eqriv 2350 1 (AB) = (AB)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ∪ cun 3207   ∩ cin 3208  ∩cint 3926 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-int 3927 This theorem is referenced by:  intunsn  3965
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