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Theorem inass 3183
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 393 . . . 4 (((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
2 elin 3156 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
32anbi2i 445 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
41, 3bitr4i 185 . . 3 (((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐶) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
5 elin 3156 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
65anbi1i 446 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐶))
7 elin 3156 . . 3 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
84, 6, 73bitr4i 210 . 2 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶)))
98ineqri 3166 1 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wcel 1434  cin 2973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980
This theorem is referenced by:  in12  3184  in32  3185  in4  3189  indif2  3215  difun1  3231  dfrab3ss  3249  resres  4652  inres  4657  imainrect  4796
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