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Theorem iotabi 5182
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 2291 . . . . . 6  |-  ( A. x ( ph  <->  ps )  <->  { x  |  ph }  =  { x  |  ps } )
21biimpi 120 . . . . 5  |-  ( A. x ( ph  <->  ps )  ->  { x  |  ph }  =  { x  |  ps } )
32eqeq1d 2186 . . . 4  |-  ( A. x ( ph  <->  ps )  ->  ( { x  | 
ph }  =  {
z }  <->  { x  |  ps }  =  {
z } ) )
43abbidv 2295 . . 3  |-  ( A. x ( ph  <->  ps )  ->  { z  |  {
x  |  ph }  =  { z } }  =  { z  |  {
x  |  ps }  =  { z } }
)
54unieqd 3818 . 2  |-  ( A. x ( ph  <->  ps )  ->  U. { z  |  { x  |  ph }  =  { z } }  =  U. { z  |  {
x  |  ps }  =  { z } }
)
6 df-iota 5173 . 2  |-  ( iota
x ph )  =  U. { z  |  {
x  |  ph }  =  { z } }
7 df-iota 5173 . 2  |-  ( iota
x ps )  = 
U. { z  |  { x  |  ps }  =  { z } }
85, 6, 73eqtr4g 2235 1  |-  ( A. x ( ph  <->  ps )  ->  ( iota x ph )  =  ( iota x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   A.wal 1351    = wceq 1353   {cab 2163   {csn 3591   U.cuni 3807   iotacio 5171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-uni 3808  df-iota 5173
This theorem is referenced by:  iotabidv  5194  iotabii  5195  eusvobj1  5855
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