Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iuneq12df Structured version   Visualization version   Unicode version

Theorem iuneq12df 4293
 Description: Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.)
Hypotheses
Ref Expression
iuneq12df.1
iuneq12df.2
iuneq12df.3
iuneq12df.4
iuneq12df.5
Assertion
Ref Expression
iuneq12df

Proof of Theorem iuneq12df
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 iuneq12df.1 . . . 4
2 iuneq12df.2 . . . 4
3 iuneq12df.3 . . . 4
4 iuneq12df.4 . . . 4
5 iuneq12df.5 . . . . 5
65eleq2d 2534 . . . 4
71, 2, 3, 4, 6rexeqbid 2986 . . 3
87alrimiv 1781 . 2
9 abbi 2585 . . 3
10 df-iun 4271 . . . 4
11 df-iun 4271 . . . 4
1210, 11eqeq12i 2485 . . 3
139, 12bitr4i 260 . 2
148, 13sylib 201 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189  wal 1450   wceq 1452  wnf 1675   wcel 1904  cab 2457  wnfc 2599  wrex 2757  ciun 4269 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-iun 4271 This theorem is referenced by:  iunxdif3  4355  iundisjf  28276  aciunf1  28340  measvuni  29110  iuneq2f  32462
 Copyright terms: Public domain W3C validator