Theorem iotaeq 4875

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 1680	. . . . . . 7 |- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) )
2		df-clab 2027	. . . . . . 7 |- ( z e. { x | ph } <-> [ z / x ] ph )
3		df-clab 2027	. . . . . . 7 |- ( z e. { y | ph } <-> [ z / y ] ph )
4	1, 2, 3	3bitr4g 212	. . . . . 6 |- ( A. x x = y -> ( z e. { x | ph } <-> z e. { y | ph } ) )
5	4	eqrdv 2038	. . . . 5 |- ( A. x x = y -> { x | ph } = { y | ph } )
6	5	eqeq1d 2048	. . . 4 |- ( A. x x = y -> ( { x | ph } = { z } <-> { y | ph } = { z } ) )
7	6	abbidv 2155	. . 3 |- ( A. x x = y -> { z | { x | ph } = { z } } = { z | { y | ph } = { z } } )
8	7	unieqd 3591	. 2 |- ( A. x x = y -> U. { z | { x | ph } = { z } } = U. { z | { y | ph } = { z } } )
9		df-iota 4867	. 2 |- ( iota x ph ) = U. { z | { x | ph } = { z } }
10		df-iota 4867	. 2 |- ( iota y ph ) = U. { z | { y | ph } = { z } }
11	8, 9, 10	3eqtr4g 2097	1 |- ( A. x x = y -> ( iota x ph ) = ( iota y ph ) )

Syntax hints:   -> wi 4   A.wal 1241   = wceq 1243   e. wcel 1393   [wsb 1645   {cab 2026   {csn 3375   U.cuni 3580   iotacio 4865

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-uni 3581  df-iota 4867

This theorem is referenced by: (None)