Theorem iotabi 4896

Description: Equivalence theorem for descriptions. (Contributed by Andrew Salmon, 30-Jun-2011.)

Assertion

Ref	Expression
iotabi	|- ( A. x ( ph <-> ps ) -> ( iota x ph ) = ( iota x ps ) )

Proof of Theorem iotabi

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		abbi 2192	. . . . . 6 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )
2	1	biimpi 118	. . . . 5 |- ( A. x ( ph <-> ps ) -> { x | ph } = { x | ps } )
3	2	eqeq1d 2089	. . . 4 |- ( A. x ( ph <-> ps ) -> ( { x | ph } = { z } <-> { x | ps } = { z } ) )
4	3	abbidv 2196	. . 3 |- ( A. x ( ph <-> ps ) -> { z | { x | ph } = { z } } = { z | { x | ps } = { z } } )
5	4	unieqd 3612	. 2 |- ( A. x ( ph <-> ps ) -> U. { z | { x | ph } = { z } } = U. { z | { x | ps } = { z } } )
6		df-iota 4887	. 2 |- ( iota x ph ) = U. { z | { x | ph } = { z } }
7		df-iota 4887	. 2 |- ( iota x ps ) = U. { z | { x | ps } = { z } }
8	5, 6, 7	3eqtr4g 2138	1 |- ( A. x ( ph <-> ps ) -> ( iota x ph ) = ( iota x ps ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   = wceq 1284   {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:  iotabidv  4908  iotabii  4909  eusvobj1  5519