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Theorem iotabi 4976
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 2201 . . . . . 6  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
21biimpi 118 . . . . 5  |-  ( A. x ( ph  <->  ps )  ->  { x  |  ph }  =  { x  |  ps } )
32eqeq1d 2096 . . . 4  |-  ( A. x ( ph  <->  ps )  ->  ( { x  | 
ph }  =  {
z }  <->  { x  |  ps }  =  {
z } ) )
43abbidv 2205 . . 3  |-  ( A. x ( ph  <->  ps )  ->  { z  |  {
x  |  ph }  =  { z } }  =  { z  |  {
x  |  ps }  =  { z } }
)
54unieqd 3659 . 2  |-  ( A. x ( ph  <->  ps )  ->  U. { z  |  { x  |  ph }  =  { z } }  =  U. { z  |  {
x  |  ps }  =  { z } }
)
6 df-iota 4967 . 2  |-  ( iota
x ph )  =  U. { z  |  {
x  |  ph }  =  { z } }
7 df-iota 4967 . 2  |-  ( iota
x ps )  = 
U. { z  |  { x  |  ps }  =  { z } }
85, 6, 73eqtr4g 2145 1  |-  ( A. x ( ph  <->  ps )  ->  ( iota x ph )  =  ( iota x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1287    = wceq 1289   {cab 2074   {csn 3441   U.cuni 3648   iotacio 4965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-uni 3649  df-iota 4967
This theorem is referenced by:  iotabidv  4988  iotabii  4989  eusvobj1  5621
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