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Theorem iuneq12daf 28160
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 2492 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
y  e.  C  <->  y  e.  D ) )
41, 3rexbida 2934 . . . 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 3022 . . . . 5  |-  ( A  =  B  ->  ( E. x  e.  A  y  e.  D  <->  E. x  e.  B  y  e.  D ) )
95, 8syl 17 . . . 4  |-  ( ph  ->  ( E. x  e.  A  y  e.  D  <->  E. x  e.  B  y  e.  D ) )
104, 9bitrd 256 . . 3  |-  ( ph  ->  ( E. x  e.  A  y  e.  C  <->  E. x  e.  B  y  e.  D ) )
1110alrimiv 1763 . 2  |-  ( ph  ->  A. y ( E. x  e.  A  y  e.  C  <->  E. x  e.  B  y  e.  D ) )
12 abbi 2553 . . 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 4298 . . . 4  |-  U_ x  e.  A  C  =  { y  |  E. x  e.  A  y  e.  C }
14 df-iun 4298 . . . 4  |-  U_ x  e.  B  D  =  { y  |  E. x  e.  B  y  e.  D }
1513, 14eqeq12i 2442 . . 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 255 . 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 199 1  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   F/wnf 1663    e. wcel 1868   {cab 2407   F/_wnfc 2570   E.wrex 2776   U_ciun 4296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-rex 2781  df-iun 4298
This theorem is referenced by:  measvunilem0  29031
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