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Theorem iotabi 5067
Description: Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
Assertion
Ref Expression
iotabi  |-  ( A. x ( ph  <->  ps )  ->  ( iota x ph )  =  ( iota x ps ) )

Proof of Theorem iotabi
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 abbi 2231 . . . . . 6  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
21biimpi 119 . . . . 5  |-  ( A. x ( ph  <->  ps )  ->  { x  |  ph }  =  { x  |  ps } )
32eqeq1d 2126 . . . 4  |-  ( A. x ( ph  <->  ps )  ->  ( { x  | 
ph }  =  {
z }  <->  { x  |  ps }  =  {
z } ) )
43abbidv 2235 . . 3  |-  ( A. x ( ph  <->  ps )  ->  { z  |  {
x  |  ph }  =  { z } }  =  { z  |  {
x  |  ps }  =  { z } }
)
54unieqd 3717 . 2  |-  ( A. x ( ph  <->  ps )  ->  U. { z  |  { x  |  ph }  =  { z } }  =  U. { z  |  {
x  |  ps }  =  { z } }
)
6 df-iota 5058 . 2  |-  ( iota
x ph )  =  U. { z  |  {
x  |  ph }  =  { z } }
7 df-iota 5058 . 2  |-  ( iota
x ps )  = 
U. { z  |  { x  |  ps }  =  { z } }
85, 6, 73eqtr4g 2175 1  |-  ( A. x ( ph  <->  ps )  ->  ( iota x ph )  =  ( iota x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1314    = wceq 1316   {cab 2103   {csn 3497   U.cuni 3706   iotacio 5056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-uni 3707  df-iota 5058
This theorem is referenced by:  iotabidv  5079  iotabii  5080  eusvobj1  5729
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