Theorem iotaeq 4895

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.

Step	Hyp	Ref	Expression
1		drsb1 1720	. . . . . . 7 |- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) )
2		df-clab 2068	. . . . . . 7 |- ( z e. { x | ph } <-> [ z / x ] ph )
3		df-clab 2068	. . . . . . 7 |- ( z e. { y | ph } <-> [ z / y ] ph )
4	1, 2, 3	3bitr4g 221	. . . . . 6 |- ( A. x x = y -> ( z e. { x | ph } <-> z e. { y | ph } ) )
5	4	eqrdv 2079	. . . . 5 |- ( A. x x = y -> { x | ph } = { y | ph } )
6	5	eqeq1d 2089	. . . 4 |- ( A. x x = y -> ( { x | ph } = { z } <-> { y | ph } = { z } ) )
7	6	abbidv 2196	. . 3 |- ( A. x x = y -> { z | { x | ph } = { z } } = { z | { y | ph } = { z } } )
8	7	unieqd 3612	. 2 |- ( A. x x = y -> U. { z | { x | ph } = { z } } = U. { z | { y | ph } = { z } } )
9		df-iota 4887	. 2 |- ( iota x ph ) = U. { z | { x | ph } = { z } }
10		df-iota 4887	. 2 |- ( iota y ph ) = U. { z | { y | ph } = { z } }
11	8, 9, 10	3eqtr4g 2138	1 |- ( A. x x = y -> ( iota x ph ) = ( iota y ph ) )

Syntax hints:   -> wi 4   A.wal 1282   = wceq 1284   e. wcel 1433   [wsb 1685   {cab 2067   {csn 3398   U.cuni 3601   iotacio 4885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-uni 3602  df-iota 4887

This theorem is referenced by: (None)