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Theorem intun 3673
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 1386 . . . 4 (∀𝑦((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)) ↔ (∀𝑦(𝑦𝐴𝑥𝑦) ∧ ∀𝑦(𝑦𝐵𝑥𝑦)))
2 elun 3111 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
32imbi1i 231 . . . . . 6 ((𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ ((𝑦𝐴𝑦𝐵) → 𝑥𝑦))
4 jaob 641 . . . . . 6 (((𝑦𝐴𝑦𝐵) → 𝑥𝑦) ↔ ((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)))
53, 4bitri 177 . . . . 5 ((𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ ((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)))
65albii 1375 . . . 4 (∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ ∀𝑦((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)))
7 vex 2577 . . . . . 6 𝑥 ∈ V
87elint 3648 . . . . 5 (𝑥 𝐴 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
97elint 3648 . . . . 5 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
108, 9anbi12i 441 . . . 4 ((𝑥 𝐴𝑥 𝐵) ↔ (∀𝑦(𝑦𝐴𝑥𝑦) ∧ ∀𝑦(𝑦𝐵𝑥𝑦)))
111, 6, 103bitr4i 205 . . 3 (∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ (𝑥 𝐴𝑥 𝐵))
127elint 3648 . . 3 (𝑥 (𝐴𝐵) ↔ ∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦))
13 elin 3153 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴𝑥 𝐵))
1411, 12, 133bitr4i 205 . 2 (𝑥 (𝐴𝐵) ↔ 𝑥 ∈ ( 𝐴 𝐵))
1514eqriv 2053 1 (𝐴𝐵) = ( 𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wo 639  wal 1257   = wceq 1259  wcel 1409  cun 2942  cin 2943   cint 3642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-int 3643
This theorem is referenced by:  intunsn  3680  riinint  4620
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