Theorem iotabi 4906

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 2193	. . . . . 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 2090	. . . 4 |- ( A. x ( ph <-> ps ) -> ( { x | ph } = { z } <-> { x | ps } = { z } ) )
4	3	abbidv 2197	. . 3 |- ( A. x ( ph <-> ps ) -> { z | { x | ph } = { z } } = { z | { x | ps } = { z } } )
5	4	unieqd 3620	. 2 |- ( A. x ( ph <-> ps ) -> U. { z | { x | ph } = { z } } = U. { z | { x | ps } = { z } } )
6		df-iota 4897	. 2 |- ( iota x ph ) = U. { z | { x | ph } = { z } }
7		df-iota 4897	. 2 |- ( iota x ps ) = U. { z | { x | ps } = { z } }
8	5, 6, 7	3eqtr4g 2139	1 |- ( A. x ( ph <-> ps ) -> ( iota x ph ) = ( iota x ps ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1283   = wceq 1285   {cab 2068   {csn 3406   U.cuni 3609   iotacio 4895

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-uni 3610  df-iota 4897

This theorem is referenced by:  iotabidv  4918  iotabii  4919  eusvobj1  5530