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Theorem iuniin 3657
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 2416 . . . 4 (x A y B z 𝐶y B x A z 𝐶)
2 vex 2554 . . . . . 6 z V
3 eliin 3652 . . . . . 6 (z V → (z y B 𝐶y B z 𝐶))
42, 3ax-mp 7 . . . . 5 (z y B 𝐶y B z 𝐶)
54rexbii 2325 . . . 4 (x A z y B 𝐶x A y B z 𝐶)
6 eliun 3651 . . . . 5 (z x A 𝐶x A z 𝐶)
76ralbii 2324 . . . 4 (y B z x A 𝐶y B x A z 𝐶)
81, 5, 73imtr4i 190 . . 3 (x A z y B 𝐶y B z x A 𝐶)
9 eliun 3651 . . 3 (z x A y B 𝐶x A z y B 𝐶)
10 eliin 3652 . . . 4 (z V → (z y B x A 𝐶y B z x A 𝐶))
112, 10ax-mp 7 . . 3 (z y B x A 𝐶y B z x A 𝐶)
128, 9, 113imtr4i 190 . 2 (z x A y B 𝐶z y B x A 𝐶)
1312ssriv 2943 1 x A y B 𝐶 y B x A 𝐶
Colors of variables: wff set class
Syntax hints:  wb 98   wcel 1390  wral 2300  wrex 2301  Vcvv 2551  wss 2911   ciun 3647   ciin 3648
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iun 3649  df-iin 3650
This theorem is referenced by: (None)
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