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Theorem inass 3211
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 394 . . . 4 (((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐶) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
2 elin 3184 . . . . 5 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
32anbi2i 446 . . . 4 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
41, 3bitr4i 186 . . 3 (((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐶) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
5 elin 3184 . . . 4 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
65anbi1i 447 . . 3 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑥𝐶))
7 elin 3184 . . 3 (𝑥 ∈ (𝐴 ∩ (𝐵𝐶)) ↔ (𝑥𝐴𝑥 ∈ (𝐵𝐶)))
84, 6, 73bitr4i 211 . 2 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑥𝐶) ↔ 𝑥 ∈ (𝐴 ∩ (𝐵𝐶)))
98ineqri 3194 1 ((𝐴𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1290  wcel 1439  cin 2999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006
This theorem is referenced by:  in12  3212  in32  3213  in4  3217  indif2  3244  difun1  3260  dfrab3ss  3278  resres  4738  inres  4743  imainrect  4889
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