Theorem iotabi 5162

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 2280	. . . . . 6 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )
2	1	biimpi 119	. . . . 5 |- ( A. x ( ph <-> ps ) -> { x | ph } = { x | ps } )
3	2	eqeq1d 2174	. . . 4 |- ( A. x ( ph <-> ps ) -> ( { x | ph } = { z } <-> { x | ps } = { z } ) )
4	3	abbidv 2284	. . 3 |- ( A. x ( ph <-> ps ) -> { z | { x | ph } = { z } } = { z | { x | ps } = { z } } )
5	4	unieqd 3800	. 2 |- ( A. x ( ph <-> ps ) -> U. { z | { x | ph } = { z } } = U. { z | { x | ps } = { z } } )
6		df-iota 5153	. 2 |- ( iota x ph ) = U. { z | { x | ph } = { z } }
7		df-iota 5153	. 2 |- ( iota x ps ) = U. { z | { x | ps } = { z } }
8	5, 6, 7	3eqtr4g 2224	1 |- ( A. x ( ph <-> ps ) -> ( iota x ph ) = ( iota x ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   {cab 2151   {csn 3576   U.cuni 3789   iotacio 5151

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-uni 3790  df-iota 5153

This theorem is referenced by:  iotabidv  5174  iotabii  5175  eusvobj1  5829