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Theorem iotaeq 5168
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 1792 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] ph  <->  [ z  /  y ] ph ) )
2 df-clab 2157 . . . . . . 7  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
3 df-clab 2157 . . . . . . 7  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
41, 2, 33bitr4g 222 . . . . . 6  |-  ( A. x  x  =  y  ->  ( z  e.  {
x  |  ph }  <->  z  e.  { y  | 
ph } ) )
54eqrdv 2168 . . . . 5  |-  ( A. x  x  =  y  ->  { x  |  ph }  =  { y  |  ph } )
65eqeq1d 2179 . . . 4  |-  ( A. x  x  =  y  ->  ( { x  | 
ph }  =  {
z }  <->  { y  |  ph }  =  {
z } ) )
76abbidv 2288 . . 3  |-  ( A. x  x  =  y  ->  { z  |  {
x  |  ph }  =  { z } }  =  { z  |  {
y  |  ph }  =  { z } }
)
87unieqd 3807 . 2  |-  ( A. x  x  =  y  ->  U. { z  |  { x  |  ph }  =  { z } }  =  U. { z  |  {
y  |  ph }  =  { z } }
)
9 df-iota 5160 . 2  |-  ( iota
x ph )  =  U. { z  |  {
x  |  ph }  =  { z } }
10 df-iota 5160 . 2  |-  ( iota y ph )  = 
U. { z  |  { y  |  ph }  =  { z } }
118, 9, 103eqtr4g 2228 1  |-  ( A. x  x  =  y  ->  ( iota x ph )  =  ( iota y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1346    = wceq 1348   [wsb 1755    e. wcel 2141   {cab 2156   {csn 3583   U.cuni 3796   iotacio 5158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-uni 3797  df-iota 5160
This theorem is referenced by: (None)
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