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Theorem intun 3770
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 1440 . . . 4 (∀𝑦((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)) ↔ (∀𝑦(𝑦𝐴𝑥𝑦) ∧ ∀𝑦(𝑦𝐵𝑥𝑦)))
2 elun 3185 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
32imbi1i 237 . . . . . 6 ((𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ ((𝑦𝐴𝑦𝐵) → 𝑥𝑦))
4 jaob 682 . . . . . 6 (((𝑦𝐴𝑦𝐵) → 𝑥𝑦) ↔ ((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)))
53, 4bitri 183 . . . . 5 ((𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ ((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)))
65albii 1429 . . . 4 (∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ ∀𝑦((𝑦𝐴𝑥𝑦) ∧ (𝑦𝐵𝑥𝑦)))
7 vex 2661 . . . . . 6 𝑥 ∈ V
87elint 3745 . . . . 5 (𝑥 𝐴 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
97elint 3745 . . . . 5 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
108, 9anbi12i 453 . . . 4 ((𝑥 𝐴𝑥 𝐵) ↔ (∀𝑦(𝑦𝐴𝑥𝑦) ∧ ∀𝑦(𝑦𝐵𝑥𝑦)))
111, 6, 103bitr4i 211 . . 3 (∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦) ↔ (𝑥 𝐴𝑥 𝐵))
127elint 3745 . . 3 (𝑥 (𝐴𝐵) ↔ ∀𝑦(𝑦 ∈ (𝐴𝐵) → 𝑥𝑦))
13 elin 3227 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴𝑥 𝐵))
1411, 12, 133bitr4i 211 . 2 (𝑥 (𝐴𝐵) ↔ 𝑥 ∈ ( 𝐴 𝐵))
1514eqriv 2112 1 (𝐴𝐵) = ( 𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 680  wal 1312   = wceq 1314  wcel 1463  cun 3037  cin 3038   cint 3739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-int 3740
This theorem is referenced by:  intunsn  3777  riinint  4768
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