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Theorem iuniin 4930
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 3324 . . . 4 (∃𝑥𝐴𝑦𝐵 𝑧𝐶 → ∀𝑦𝐵𝑥𝐴 𝑧𝐶)
2 eliin 4923 . . . . . 6 (𝑧 ∈ V → (𝑧 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 𝑧𝐶))
32elv 3499 . . . . 5 (𝑧 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 𝑧𝐶)
43rexbii 3247 . . . 4 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝑧𝐶)
5 eliun 4922 . . . . 5 (𝑧 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑧𝐶)
65ralbii 3165 . . . 4 (∀𝑦𝐵 𝑧 𝑥𝐴 𝐶 ↔ ∀𝑦𝐵𝑥𝐴 𝑧𝐶)
71, 4, 63imtr4i 294 . . 3 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 → ∀𝑦𝐵 𝑧 𝑥𝐴 𝐶)
8 eliun 4922 . . 3 (𝑧 𝑥𝐴 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴 𝑧 𝑦𝐵 𝐶)
9 eliin 4923 . . . 4 (𝑧 ∈ V → (𝑧 𝑦𝐵 𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑧 𝑥𝐴 𝐶))
109elv 3499 . . 3 (𝑧 𝑦𝐵 𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑧 𝑥𝐴 𝐶)
117, 8, 103imtr4i 294 . 2 (𝑧 𝑥𝐴 𝑦𝐵 𝐶𝑧 𝑦𝐵 𝑥𝐴 𝐶)
1211ssriv 3970 1 𝑥𝐴 𝑦𝐵 𝐶 𝑦𝐵 𝑥𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wb 208  wcel 2110  wral 3138  wrex 3139  Vcvv 3494  wss 3935   ciun 4918   ciin 4919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-in 3942  df-ss 3951  df-iun 4920  df-iin 4921
This theorem is referenced by: (None)
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