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Theorem iotabi 5427
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 2546 . . . . . 6  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
21biimpi 187 . . . . 5  |-  ( A. x ( ph  <->  ps )  ->  { x  |  ph }  =  { x  |  ps } )
32eqeq1d 2444 . . . 4  |-  ( A. x ( ph  <->  ps )  ->  ( { x  | 
ph }  =  {
z }  <->  { x  |  ps }  =  {
z } ) )
43abbidv 2550 . . 3  |-  ( A. x ( ph  <->  ps )  ->  { z  |  {
x  |  ph }  =  { z } }  =  { z  |  {
x  |  ps }  =  { z } }
)
54unieqd 4026 . 2  |-  ( A. x ( ph  <->  ps )  ->  U. { z  |  { x  |  ph }  =  { z } }  =  U. { z  |  {
x  |  ps }  =  { z } }
)
6 df-iota 5418 . 2  |-  ( iota
x ph )  =  U. { z  |  {
x  |  ph }  =  { z } }
7 df-iota 5418 . 2  |-  ( iota
x ps )  = 
U. { z  |  { x  |  ps }  =  { z } }
85, 6, 73eqtr4g 2493 1  |-  ( A. x ( ph  <->  ps )  ->  ( iota x ph )  =  ( iota x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549    = wceq 1652   {cab 2422   {csn 3814   U.cuni 4015   iotacio 5416
This theorem is referenced by:  iotabidv  5439  iotabii  5440  eusvobj1  6583  iotasbcq  27614
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-uni 4016  df-iota 5418
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