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Theorem iotabi 5566
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 2598 . . . . . 6  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
21biimpi 194 . . . . 5  |-  ( A. x ( ph  <->  ps )  ->  { x  |  ph }  =  { x  |  ps } )
32eqeq1d 2469 . . . 4  |-  ( A. x ( ph  <->  ps )  ->  ( { x  | 
ph }  =  {
z }  <->  { x  |  ps }  =  {
z } ) )
43abbidv 2603 . . 3  |-  ( A. x ( ph  <->  ps )  ->  { z  |  {
x  |  ph }  =  { z } }  =  { z  |  {
x  |  ps }  =  { z } }
)
54unieqd 4261 . 2  |-  ( A. x ( ph  <->  ps )  ->  U. { z  |  { x  |  ph }  =  { z } }  =  U. { z  |  {
x  |  ps }  =  { z } }
)
6 df-iota 5557 . 2  |-  ( iota
x ph )  =  U. { z  |  {
x  |  ph }  =  { z } }
7 df-iota 5557 . 2  |-  ( iota
x ps )  = 
U. { z  |  { x  |  ps }  =  { z } }
85, 6, 73eqtr4g 2533 1  |-  ( A. x ( ph  <->  ps )  ->  ( iota x ph )  =  ( iota x ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377    = wceq 1379   {cab 2452   {csn 4033   U.cuni 4251   iotacio 5555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2823  df-uni 4252  df-iota 5557
This theorem is referenced by:  iotabidv  5578  iotabii  5579  eusvobj1  6289  iotasbcq  31246  fourierdlem89  31819  fourierdlem90  31820  fourierdlem91  31821  fourierdlem96  31826  fourierdlem97  31827  fourierdlem98  31828  fourierdlem99  31829  fourierdlem100  31830  fourierdlem104  31834  fourierdlem112  31842
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