Theorem eliminable-abeqab

Description: A theorem used to prove the base case of the Eliminability Theorem (see section comment): abstraction equals abstraction. (Contributed by BJ, 30-Apr-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	eliminable-abeqab	|- ( { x | ph } = { y | ps } <-> A. z ( [ z / x ] ph <-> [ z / y ] ps ) )

Proof

Step	Hyp	Ref	Expression
1		dfcleq	|- ( { x | ph } = { y | ps } <-> A. z ( z e. { x | ph } <-> z e. { y | ps } ) )
2		eliminable-velab	|- ( z e. { x | ph } <-> [ z / x ] ph )
3		eliminable-velab	|- ( z e. { y | ps } <-> [ z / y ] ps )
4	2 3	bibi12i	|- ( ( z e. { x | ph } <-> z e. { y | ps } ) <-> ( [ z / x ] ph <-> [ z / y ] ps ) )
5	4	albii	|- ( A. z ( z e. { x | ph } <-> z e. { y | ps } ) <-> A. z ( [ z / x ] ph <-> [ z / y ] ps ) )
6	1 5	bitri	|- ( { x | ph } = { y | ps } <-> A. z ( [ z / x ] ph <-> [ z / y ] ps ) )