Theorem iotaeq 4975

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 1727	. . . . . . 7 |- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) )
2		df-clab 2075	. . . . . . 7 |- ( z e. { x | ph } <-> [ z / x ] ph )
3		df-clab 2075	. . . . . . 7 |- ( z e. { y | ph } <-> [ z / y ] ph )
4	1, 2, 3	3bitr4g 221	. . . . . 6 |- ( A. x x = y -> ( z e. { x | ph } <-> z e. { y | ph } ) )
5	4	eqrdv 2086	. . . . 5 |- ( A. x x = y -> { x | ph } = { y | ph } )
6	5	eqeq1d 2096	. . . 4 |- ( A. x x = y -> ( { x | ph } = { z } <-> { y | ph } = { z } ) )
7	6	abbidv 2205	. . 3 |- ( A. x x = y -> { z | { x | ph } = { z } } = { z | { y | ph } = { z } } )
8	7	unieqd 3659	. 2 |- ( A. x x = y -> U. { z | { x | ph } = { z } } = U. { z | { y | ph } = { z } } )
9		df-iota 4967	. 2 |- ( iota x ph ) = U. { z | { x | ph } = { z } }
10		df-iota 4967	. 2 |- ( iota y ph ) = U. { z | { y | ph } = { z } }
11	8, 9, 10	3eqtr4g 2145	1 |- ( A. x x = y -> ( iota x ph ) = ( iota y ph ) )

Syntax hints:   -> wi 4   A.wal 1287   = wceq 1289   e. wcel 1438   [wsb 1692   {cab 2074   {csn 3441   U.cuni 3648   iotacio 4965

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-uni 3649  df-iota 4967

This theorem is referenced by: (None)