Theorem iotaeq

Description: Equality theorem for descriptions. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Andrew Salmon, 30-Jun-2011) (New usage is discouraged.)

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

Proof

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