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Theorem iuneq12daf 23844
Description: Equality deduction for indexed union, deduction version. (Contributed by Thierry Arnoux, 13-Mar-2017.)
Hypotheses
Ref Expression
iuneq12daf.1  |-  F/ x ph
iuneq12daf.2  |-  F/_ x A
iuneq12daf.3  |-  F/_ x B
iuneq12daf.4  |-  ( ph  ->  A  =  B )
iuneq12daf.5  |-  ( (
ph  /\  x  e.  A )  ->  C  =  D )
Assertion
Ref Expression
iuneq12daf  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )

Proof of Theorem iuneq12daf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 iuneq12daf.1 . . . . 5  |-  F/ x ph
2 iuneq12daf.5 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  C  =  D )
32eleq2d 2447 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
y  e.  C  <->  y  e.  D ) )
41, 3rexbida 2657 . . . 4  |-  ( ph  ->  ( E. x  e.  A  y  e.  C  <->  E. x  e.  A  y  e.  D ) )
5 iuneq12daf.4 . . . . 5  |-  ( ph  ->  A  =  B )
6 iuneq12daf.2 . . . . . 6  |-  F/_ x A
7 iuneq12daf.3 . . . . . 6  |-  F/_ x B
86, 7rexeqf 2837 . . . . 5  |-  ( A  =  B  ->  ( E. x  e.  A  y  e.  D  <->  E. x  e.  B  y  e.  D ) )
95, 8syl 16 . . . 4  |-  ( ph  ->  ( E. x  e.  A  y  e.  D  <->  E. x  e.  B  y  e.  D ) )
104, 9bitrd 245 . . 3  |-  ( ph  ->  ( E. x  e.  A  y  e.  C  <->  E. x  e.  B  y  e.  D ) )
1110alrimiv 1638 . 2  |-  ( ph  ->  A. y ( E. x  e.  A  y  e.  C  <->  E. x  e.  B  y  e.  D ) )
12 abbi 2490 . . 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 } )
13 df-iun 4030 . . . 4  |-  U_ x  e.  A  C  =  { y  |  E. x  e.  A  y  e.  C }
14 df-iun 4030 . . . 4  |-  U_ x  e.  B  D  =  { y  |  E. x  e.  B  y  e.  D }
1513, 14eqeq12i 2393 . . 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 } )
1612, 15bitr4i 244 . 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 )
1711, 16sylib 189 1  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   F/wnf 1550    = wceq 1649    e. wcel 1717   {cab 2366   F/_wnfc 2503   E.wrex 2643   U_ciun 4028
This theorem is referenced by:  measvunilem0  24354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rex 2648  df-iun 4030
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