Theorem iotabi 5303

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 2345	. . . . . 6 |- ( A. x ( ph <-> ps ) <-> { x | ph } = { x | ps } )
2	1	biimpi 120	. . . . 5 |- ( A. x ( ph <-> ps ) -> { x | ph } = { x | ps } )
3	2	eqeq1d 2240	. . . 4 |- ( A. x ( ph <-> ps ) -> ( { x | ph } = { z } <-> { x | ps } = { z } ) )
4	3	abbidv 2350	. . 3 |- ( A. x ( ph <-> ps ) -> { z | { x | ph } = { z } } = { z | { x | ps } = { z } } )
5	4	unieqd 3909	. 2 |- ( A. x ( ph <-> ps ) -> U. { z | { x | ph } = { z } } = U. { z | { x | ps } = { z } } )
6		df-iota 5293	. 2 |- ( iota x ph ) = U. { z | { x | ph } = { z } }
7		df-iota 5293	. 2 |- ( iota x ps ) = U. { z | { x | ps } = { z } }
8	5, 6, 7	3eqtr4g 2289	1 |- ( A. x ( ph <-> ps ) -> ( iota x ph ) = ( iota x ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1396   = wceq 1398   {cab 2217   {csn 3673   U.cuni 3898   iotacio 5291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-uni 3899  df-iota 5293

This theorem is referenced by:  iotabidv  5316  iotabii  5317  iotaexel  5986  eusvobj1  6015