Theorem iotaeq 5161

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

Assertion

Ref	Expression
iotaeq	|- ( A. x x = y -> ( iota x ph ) = ( iota y ph ) )

Proof of Theorem iotaeq

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		drsb1 1787	. . . . . . 7 |- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) )
2		df-clab 2152	. . . . . . 7 |- ( z e. { x | ph } <-> [ z / x ] ph )
3		df-clab 2152	. . . . . . 7 |- ( z e. { y | ph } <-> [ z / y ] ph )
4	1, 2, 3	3bitr4g 222	. . . . . 6 |- ( A. x x = y -> ( z e. { x | ph } <-> z e. { y | ph } ) )
5	4	eqrdv 2163	. . . . 5 |- ( A. x x = y -> { x | ph } = { y | ph } )
6	5	eqeq1d 2174	. . . 4 |- ( A. x x = y -> ( { x | ph } = { z } <-> { y | ph } = { z } ) )
7	6	abbidv 2284	. . 3 |- ( A. x x = y -> { z | { x | ph } = { z } } = { z | { y | ph } = { z } } )
8	7	unieqd 3800	. 2 |- ( A. x x = y -> U. { z | { x | ph } = { z } } = U. { z | { y | ph } = { z } } )
9		df-iota 5153	. 2 |- ( iota x ph ) = U. { z | { x | ph } = { z } }
10		df-iota 5153	. 2 |- ( iota y ph ) = U. { z | { y | ph } = { z } }
11	8, 9, 10	3eqtr4g 2224	1 |- ( A. x x = y -> ( iota x ph ) = ( iota y ph ) )

Syntax hints:   -> wi 4   A.wal 1341   = wceq 1343   [wsb 1750   e. wcel 2136   {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: (None)