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Theorem intun 4444
 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 1786 . . . 4 (∀𝑦((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)) ↔ (∀𝑦(𝑦𝐴𝑥𝑦) ∧ ∀𝑦(𝑦𝐵𝑥𝑦)))
2 elun 3715 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
32imbi1i 338 . . . . . 6 ((𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ ((𝑦𝐴𝑦𝐵) → 𝑥𝑦))
4 jaob 818 . . . . . 6 (((𝑦𝐴𝑦𝐵) → 𝑥𝑦) ↔ ((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)))
53, 4bitri 263 . . . . 5 ((𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ ((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)))
65albii 1737 . . . 4 (∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ ∀𝑦((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)))
7 vex 3176 . . . . . 6 𝑥 ∈ V
87elint 4416 . . . . 5 (𝑥 𝐴 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
97elint 4416 . . . . 5 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
108, 9anbi12i 729 . . . 4 ((𝑥 𝐴𝑥 𝐵) ↔ (∀𝑦(𝑦𝐴𝑥𝑦) ∧ ∀𝑦(𝑦𝐵𝑥𝑦)))
111, 6, 103bitr4i 291 . . 3 (∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ (𝑥 𝐴𝑥 𝐵))
127elint 4416 . . 3 (𝑥 (𝐴𝐵) ↔ ∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦))
13 elin 3758 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴𝑥 𝐵))
1411, 12, 133bitr4i 291 . 2 (𝑥 (𝐴𝐵) ↔ 𝑥 ∈ ( 𝐴 𝐵))
1514eqriv 2607 1 (𝐴𝐵) = ( 𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ∩ cin 3539  ∩ cint 4410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547  df-int 4411 This theorem is referenced by:  intunsn  4451  riinint  5303  fiin  8211  elfiun  8219  elrfi  36275
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