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Theorem iuniin 3979
 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 x A y B C y B x A 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 2727 . . . 4 (x A y B z Cy B x A z C)
2 vex 2862 . . . . . 6 z V
3 eliin 3974 . . . . . 6 (z V → (z y B Cy B z C))
42, 3ax-mp 5 . . . . 5 (z y B Cy B z C)
54rexbii 2639 . . . 4 (x A z y B Cx A y B z C)
6 eliun 3973 . . . . 5 (z x A Cx A z C)
76ralbii 2638 . . . 4 (y B z x A Cy B x A z C)
81, 5, 73imtr4i 257 . . 3 (x A z y B Cy B z x A C)
9 eliun 3973 . . 3 (z x A y B Cx A z y B C)
10 eliin 3974 . . . 4 (z V → (z y B x A Cy B z x A C))
112, 10ax-mp 5 . . 3 (z y B x A Cy B z x A C)
128, 9, 113imtr4i 257 . 2 (z x A y B Cz y B x A C)
1312ssriv 3277 1 x A y B C y B x A C
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257  ∪ciun 3969  ∩ciin 3970 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iun 3971  df-iin 3972 This theorem is referenced by: (None)
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