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Theorem iuneq12df 4293
Description: Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 31-Dec-2016.)
Hypotheses
Ref Expression
iuneq12df.1  |-  F/ x ph
iuneq12df.2  |-  F/_ x A
iuneq12df.3  |-  F/_ x B
iuneq12df.4  |-  ( ph  ->  A  =  B )
iuneq12df.5  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
iuneq12df  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )

Proof of Theorem iuneq12df
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 iuneq12df.1 . . . 4  |-  F/ x ph
2 iuneq12df.2 . . . 4  |-  F/_ x A
3 iuneq12df.3 . . . 4  |-  F/_ x B
4 iuneq12df.4 . . . 4  |-  ( ph  ->  A  =  B )
5 iuneq12df.5 . . . . 5  |-  ( ph  ->  C  =  D )
65eleq2d 2534 . . . 4  |-  ( ph  ->  ( y  e.  C  <->  y  e.  D ) )
71, 2, 3, 4, 6rexeqbid 2986 . . 3  |-  ( ph  ->  ( E. x  e.  A  y  e.  C  <->  E. x  e.  B  y  e.  D ) )
87alrimiv 1781 . 2  |-  ( ph  ->  A. y ( E. x  e.  A  y  e.  C  <->  E. x  e.  B  y  e.  D ) )
9 abbi 2585 . . 3  |-  ( A. y ( E. x  e.  A  y  e.  C 
<->  E. x  e.  B  y  e.  D )  <->  { y  |  E. x  e.  A  y  e.  C }  =  {
y  |  E. x  e.  B  y  e.  D } )
10 df-iun 4271 . . . 4  |-  U_ x  e.  A  C  =  { y  |  E. x  e.  A  y  e.  C }
11 df-iun 4271 . . . 4  |-  U_ x  e.  B  D  =  { y  |  E. x  e.  B  y  e.  D }
1210, 11eqeq12i 2485 . . 3  |-  ( U_ x  e.  A  C  =  U_ x  e.  B  D 
<->  { y  |  E. x  e.  A  y  e.  C }  =  {
y  |  E. x  e.  B  y  e.  D } )
139, 12bitr4i 260 . 2  |-  ( A. y ( E. x  e.  A  y  e.  C 
<->  E. x  e.  B  y  e.  D )  <->  U_ x  e.  A  C  =  U_ x  e.  B  D )
148, 13sylib 201 1  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1450    = wceq 1452   F/wnf 1675    e. wcel 1904   {cab 2457   F/_wnfc 2599   E.wrex 2757   U_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
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