Theorem iotabi 5097

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 2253	. . . . . 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 2148	. . . 4 |- ( A. x ( ph <-> ps ) -> ( { x | ph } = { z } <-> { x | ps } = { z } ) )
4	3	abbidv 2257	. . 3 |- ( A. x ( ph <-> ps ) -> { z | { x | ph } = { z } } = { z | { x | ps } = { z } } )
5	4	unieqd 3747	. 2 |- ( A. x ( ph <-> ps ) -> U. { z | { x | ph } = { z } } = U. { z | { x | ps } = { z } } )
6		df-iota 5088	. 2 |- ( iota x ph ) = U. { z | { x | ph } = { z } }
7		df-iota 5088	. 2 |- ( iota x ps ) = U. { z | { x | ps } = { z } }
8	5, 6, 7	3eqtr4g 2197	1 |- ( A. x ( ph <-> ps ) -> ( iota x ph ) = ( iota x ps ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1329   = wceq 1331   {cab 2125   {csn 3527   U.cuni 3736   iotacio 5086

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-uni 3737  df-iota 5088

This theorem is referenced by:  iotabidv  5109  iotabii  5110  eusvobj1  5761