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Theorem iuneq12daf 23170
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 2363 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (
y  e.  C  <->  y  e.  D ) )
41, 3rexbida 2571 . . . 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 2746 . . . . 5  |-  ( A  =  B  ->  ( E. x  e.  A  y  e.  D  <->  E. x  e.  B  y  e.  D ) )
95, 8syl 15 . . . 4  |-  ( ph  ->  ( E. x  e.  A  y  e.  D  <->  E. x  e.  B  y  e.  D ) )
104, 9bitrd 244 . . 3  |-  ( ph  ->  ( E. x  e.  A  y  e.  C  <->  E. x  e.  B  y  e.  D ) )
1110alrimiv 1621 . 2  |-  ( ph  ->  A. y ( E. x  e.  A  y  e.  C  <->  E. x  e.  B  y  e.  D ) )
12 abbi 2406 . . 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 3923 . . . 4  |-  U_ x  e.  A  C  =  { y  |  E. x  e.  A  y  e.  C }
14 df-iun 3923 . . . 4  |-  U_ x  e.  B  D  =  { y  |  E. x  e.  B  y  e.  D }
1513, 14eqeq12i 2309 . . 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 243 . 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 188 1  |-  ( ph  ->  U_ x  e.  A  C  =  U_ x  e.  B  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   F/wnf 1534    = wceq 1632    e. wcel 1696   {cab 2282   F/_wnfc 2419   E.wrex 2557   U_ciun 3921
This theorem is referenced by:  measvunilem0  23556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-iun 3923
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