Theorem iotaeq 5143

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 1779	. . . . . . 7 |- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) )
2		df-clab 2144	. . . . . . 7 |- ( z e. { x | ph } <-> [ z / x ] ph )
3		df-clab 2144	. . . . . . 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 2155	. . . . 5 |- ( A. x x = y -> { x | ph } = { y | ph } )
6	5	eqeq1d 2166	. . . 4 |- ( A. x x = y -> ( { x | ph } = { z } <-> { y | ph } = { z } ) )
7	6	abbidv 2275	. . 3 |- ( A. x x = y -> { z | { x | ph } = { z } } = { z | { y | ph } = { z } } )
8	7	unieqd 3783	. 2 |- ( A. x x = y -> U. { z | { x | ph } = { z } } = U. { z | { y | ph } = { z } } )
9		df-iota 5135	. 2 |- ( iota x ph ) = U. { z | { x | ph } = { z } }
10		df-iota 5135	. 2 |- ( iota y ph ) = U. { z | { y | ph } = { z } }
11	8, 9, 10	3eqtr4g 2215	1 |- ( A. x x = y -> ( iota x ph ) = ( iota y ph ) )

Syntax hints:   -> wi 4   A.wal 1333   = wceq 1335   [wsb 1742   e. wcel 2128   {cab 2143   {csn 3560   U.cuni 3772   iotacio 5133

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-uni 3773  df-iota 5135

This theorem is referenced by: (None)