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Theorem intun 3637
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 1367 . . . 4 (y((y Ax y) (y Bx y)) ↔ (y(y Ax y) y(y Bx y)))
2 elun 3078 . . . . . . 7 (y (AB) ↔ (y A y B))
32imbi1i 227 . . . . . 6 ((y (AB) → x y) ↔ ((y A y B) → x y))
4 jaob 630 . . . . . 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 1356 . . . 4 (y(y (AB) → x y) ↔ y((y Ax y) (y Bx y)))
7 vex 2554 . . . . . 6 x V
87elint 3612 . . . . 5 (x Ay(y Ax y))
97elint 3612 . . . . 5 (x By(y Bx y))
108, 9anbi12i 433 . . . 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 3612 . . 3 (x (AB) ↔ y(y (AB) → x y))
13 elin 3120 . . 3 (x ( A B) ↔ (x A x B))
1411, 12, 133bitr4i 201 . 2 (x (AB) ↔ x ( A B))
1514eqriv 2034 1 (AB) = ( A B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 628  wal 1240   = wceq 1242   wcel 1390  cun 2909  cin 2910   cint 3606
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-bnd 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-v 2553  df-un 2916  df-in 2918  df-int 3607
This theorem is referenced by:  intunsn  3644  riinint  4536
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