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Theorem iotaeq 5426
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 2112 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] ph  <->  [ z  /  y ] ph ) )
2 df-clab 2423 . . . . . . 7  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
3 df-clab 2423 . . . . . . 7  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
41, 2, 33bitr4g 280 . . . . . 6  |-  ( A. x  x  =  y  ->  ( z  e.  {
x  |  ph }  <->  z  e.  { y  | 
ph } ) )
54eqrdv 2434 . . . . 5  |-  ( A. x  x  =  y  ->  { x  |  ph }  =  { y  |  ph } )
65eqeq1d 2444 . . . 4  |-  ( A. x  x  =  y  ->  ( { x  | 
ph }  =  {
z }  <->  { y  |  ph }  =  {
z } ) )
76abbidv 2550 . . 3  |-  ( A. x  x  =  y  ->  { z  |  {
x  |  ph }  =  { z } }  =  { z  |  {
y  |  ph }  =  { z } }
)
87unieqd 4026 . 2  |-  ( A. x  x  =  y  ->  U. { z  |  { x  |  ph }  =  { z } }  =  U. { z  |  {
y  |  ph }  =  { z } }
)
9 df-iota 5418 . 2  |-  ( iota
x ph )  =  U. { z  |  {
x  |  ph }  =  { z } }
10 df-iota 5418 . 2  |-  ( iota y ph )  = 
U. { z  |  { y  |  ph }  =  { z } }
118, 9, 103eqtr4g 2493 1  |-  ( A. x  x  =  y  ->  ( iota x ph )  =  ( iota y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549    = wceq 1652   [wsb 1658    e. wcel 1725   {cab 2422   {csn 3814   U.cuni 4015   iotacio 5416
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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