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Theorem iotaeq 4903
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 1696 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] ph  <->  [ z  /  y ] ph ) )
2 df-clab 2043 . . . . . . 7  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
3 df-clab 2043 . . . . . . 7  |-  ( z  e.  { y  | 
ph }  <->  [ z  /  y ] ph )
41, 2, 33bitr4g 216 . . . . . 6  |-  ( A. x  x  =  y  ->  ( z  e.  {
x  |  ph }  <->  z  e.  { y  | 
ph } ) )
54eqrdv 2054 . . . . 5  |-  ( A. x  x  =  y  ->  { x  |  ph }  =  { y  |  ph } )
65eqeq1d 2064 . . . 4  |-  ( A. x  x  =  y  ->  ( { x  | 
ph }  =  {
z }  <->  { y  |  ph }  =  {
z } ) )
76abbidv 2171 . . 3  |-  ( A. x  x  =  y  ->  { z  |  {
x  |  ph }  =  { z } }  =  { z  |  {
y  |  ph }  =  { z } }
)
87unieqd 3619 . 2  |-  ( A. x  x  =  y  ->  U. { z  |  { x  |  ph }  =  { z } }  =  U. { z  |  {
y  |  ph }  =  { z } }
)
9 df-iota 4895 . 2  |-  ( iota
x ph )  =  U. { z  |  {
x  |  ph }  =  { z } }
10 df-iota 4895 . 2  |-  ( iota y ph )  = 
U. { z  |  { y  |  ph }  =  { z } }
118, 9, 103eqtr4g 2113 1  |-  ( A. x  x  =  y  ->  ( iota x ph )  =  ( iota y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257    = wceq 1259    e. wcel 1409   [wsb 1661   {cab 2042   {csn 3403   U.cuni 3608   iotacio 4893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-uni 3609  df-iota 4895
This theorem is referenced by: (None)
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