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Theorem inass 3465
 Description: Associative law for intersection of classes. Exercise 9 of [TakeutiZaring] p. 17. (Contributed by NM, 3-May-1994.)
Assertion
Ref Expression
inass ((AB) ∩ C) = (A ∩ (BC))

Proof of Theorem inass
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 anass 630 . . . 4 (((x A x B) x C) ↔ (x A (x B x C)))
2 elin 3219 . . . . 5 (x (BC) ↔ (x B x C))
32anbi2i 675 . . . 4 ((x A x (BC)) ↔ (x A (x B x C)))
41, 3bitr4i 243 . . 3 (((x A x B) x C) ↔ (x A x (BC)))
5 elin 3219 . . . 4 (x (AB) ↔ (x A x B))
65anbi1i 676 . . 3 ((x (AB) x C) ↔ ((x A x B) x C))
7 elin 3219 . . 3 (x (A ∩ (BC)) ↔ (x A x (BC)))
84, 6, 73bitr4i 268 . 2 ((x (AB) x C) ↔ x (A ∩ (BC)))
98ineqri 3449 1 ((AB) ∩ C) = (A ∩ (BC))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∩ cin 3208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 This theorem is referenced by:  in12  3466  in32  3467  in4  3471  indif2  3498  difun1  3514  dfrab3ss  3533  dfif4  3673  ssfin  4470  resres  4980  inres  4985
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