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Theorem iotaeq 5565
Description: Equality theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)
Assertion
Ref Expression
iotaeq  |-  ( A. x  x  =  y  ->  ( iota x ph )  =  ( iota y ph ) )

Proof of Theorem iotaeq
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 drsb1 2091 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] ph  <->  [ z  /  y ] ph ) )
2 df-clab 2453 . . . . . . 7  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
3 df-clab 2453 . . . . . . 7  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
41, 2, 33bitr4g 288 . . . . . 6  |-  ( A. x  x  =  y  ->  ( z  e.  {
x  |  ph }  <->  z  e.  { y  | 
ph } ) )
54eqrdv 2464 . . . . 5  |-  ( A. x  x  =  y  ->  { x  |  ph }  =  { y  |  ph } )
65eqeq1d 2469 . . . 4  |-  ( A. x  x  =  y  ->  ( { x  | 
ph }  =  {
z }  <->  { y  |  ph }  =  {
z } ) )
76abbidv 2603 . . 3  |-  ( A. x  x  =  y  ->  { z  |  {
x  |  ph }  =  { z } }  =  { z  |  {
y  |  ph }  =  { z } }
)
87unieqd 4261 . 2  |-  ( A. x  x  =  y  ->  U. { z  |  { x  |  ph }  =  { z } }  =  U. { z  |  {
y  |  ph }  =  { z } }
)
9 df-iota 5557 . 2  |-  ( iota
x ph )  =  U. { z  |  {
x  |  ph }  =  { z } }
10 df-iota 5557 . 2  |-  ( iota y ph )  = 
U. { z  |  { y  |  ph }  =  { z } }
118, 9, 103eqtr4g 2533 1  |-  ( A. x  x  =  y  ->  ( iota x ph )  =  ( iota y ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1377    = wceq 1379   [wsb 1711    e. wcel 1767   {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: (None)
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