New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  iunxiun GIF version

Theorem iunxiun 4048
 Description: Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)
Assertion
Ref Expression
iunxiun x y A BC = y A x B C
Distinct variable groups:   x,y   x,A   y,C
Allowed substitution hints:   A(y)   B(x,y)   C(x)

Proof of Theorem iunxiun
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 eliun 3973 . . . . . . . 8 (x y A By A x B)
21anbi1i 676 . . . . . . 7 ((x y A B z C) ↔ (y A x B z C))
3 r19.41v 2764 . . . . . . 7 (y A (x B z C) ↔ (y A x B z C))
42, 3bitr4i 243 . . . . . 6 ((x y A B z C) ↔ y A (x B z C))
54exbii 1582 . . . . 5 (x(x y A B z C) ↔ xy A (x B z C))
6 rexcom4 2878 . . . . 5 (y A x(x B z C) ↔ xy A (x B z C))
75, 6bitr4i 243 . . . 4 (x(x y A B z C) ↔ y A x(x B z C))
8 df-rex 2620 . . . 4 (x y A Bz Cx(x y A B z C))
9 eliun 3973 . . . . . 6 (z x B Cx B z C)
10 df-rex 2620 . . . . . 6 (x B z Cx(x B z C))
119, 10bitri 240 . . . . 5 (z x B Cx(x B z C))
1211rexbii 2639 . . . 4 (y A z x B Cy A x(x B z C))
137, 8, 123bitr4i 268 . . 3 (x y A Bz Cy A z x B C)
14 eliun 3973 . . 3 (z x y A BCx y A Bz C)
15 eliun 3973 . . 3 (z y A x B Cy A z x B C)
1613, 14, 153bitr4i 268 . 2 (z x y A BCz y A x B C)
1716eqriv 2350 1 x y A BC = y A x B C
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  ∪ciun 3969 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-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-iun 3971 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator