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Theorem iotaeq 5064
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 1753 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] ph  <->  [ z  /  y ] ph ) )
2 df-clab 2102 . . . . . . 7  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
3 df-clab 2102 . . . . . . 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 2113 . . . . 5  |-  ( A. x  x  =  y  ->  { x  |  ph }  =  { y  |  ph } )
65eqeq1d 2124 . . . 4  |-  ( A. x  x  =  y  ->  ( { x  | 
ph }  =  {
z }  <->  { y  |  ph }  =  {
z } ) )
76abbidv 2233 . . 3  |-  ( A. x  x  =  y  ->  { z  |  {
x  |  ph }  =  { z } }  =  { z  |  {
y  |  ph }  =  { z } }
)
87unieqd 3715 . 2  |-  ( A. x  x  =  y  ->  U. { z  |  { x  |  ph }  =  { z } }  =  U. { z  |  {
y  |  ph }  =  { z } }
)
9 df-iota 5056 . 2  |-  ( iota
x ph )  =  U. { z  |  {
x  |  ph }  =  { z } }
10 df-iota 5056 . 2  |-  ( iota y ph )  = 
U. { z  |  { y  |  ph }  =  { z } }
118, 9, 103eqtr4g 2173 1  |-  ( A. x  x  =  y  ->  ( iota x ph )  =  ( iota y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1312    = wceq 1314    e. wcel 1463   [wsb 1718   {cab 2101   {csn 3495   U.cuni 3704   iotacio 5054
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-uni 3705  df-iota 5056
This theorem is referenced by: (None)
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