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Theorem iuniin 3667
 Description: Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iuniin 𝑥𝐴 𝑦𝐵 𝐶 𝑦𝐵 𝑥𝐴 𝐶
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem iuniin
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 r19.12 2422 . . . 4 (∃𝑥𝐴𝑦𝐵 𝑧𝐶 → ∀𝑦𝐵𝑥𝐴 𝑧𝐶)
2 vex 2560 . . . . . 6 𝑧 ∈ V
3 eliin 3662 . . . . . 6 (𝑧 ∈ V → (𝑧 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 𝑧𝐶))
42, 3ax-mp 7 . . . . 5 (𝑧 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 𝑧𝐶)
54rexbii 2331 . . . 4 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝑧𝐶)
6 eliun 3661 . . . . 5 (𝑧 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑧𝐶)
76ralbii 2330 . . . 4 (∀𝑦𝐵 𝑧 𝑥𝐴 𝐶 ↔ ∀𝑦𝐵𝑥𝐴 𝑧𝐶)
81, 5, 73imtr4i 190 . . 3 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 → ∀𝑦𝐵 𝑧 𝑥𝐴 𝐶)
9 eliun 3661 . . 3 (𝑧 𝑥𝐴 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴 𝑧 𝑦𝐵 𝐶)
10 eliin 3662 . . . 4 (𝑧 ∈ V → (𝑧 𝑦𝐵 𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑧 𝑥𝐴 𝐶))
112, 10ax-mp 7 . . 3 (𝑧 𝑦𝐵 𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑧 𝑥𝐴 𝐶)
128, 9, 113imtr4i 190 . 2 (𝑧 𝑥𝐴 𝑦𝐵 𝐶𝑧 𝑦𝐵 𝑥𝐴 𝐶)
1312ssriv 2949 1 𝑥𝐴 𝑦𝐵 𝐶 𝑦𝐵 𝑥𝐴 𝐶
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  Vcvv 2557   ⊆ wss 2917  ∪ ciun 3657  ∩ ciin 3658 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-iun 3659  df-iin 3660 This theorem is referenced by: (None)
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