Theorem eliminable-abelab

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

	Ref	Expression
Assertion	eliminable-abelab	|- ( { x | ph } e. { y | ps } <-> E. z ( A. t ( t e. z <-> [ t / x ] ph ) /\ [ z / y ] ps ) )

Proof

Step	Hyp	Ref	Expression
1		dfclel	|- ( { x | ph } e. { y | ps } <-> E. z ( z = { x | ph } /\ z e. { y | ps } ) )
2		eliminable-veqab	|- ( z = { x | ph } <-> A. t ( t e. z <-> [ t / x ] ph ) )
3		eliminable-velab	|- ( z e. { y | ps } <-> [ z / y ] ps )
4	2 3	anbi12i	|- ( ( z = { x | ph } /\ z e. { y | ps } ) <-> ( A. t ( t e. z <-> [ t / x ] ph ) /\ [ z / y ] ps ) )
5	4	exbii	|- ( E. z ( z = { x | ph } /\ z e. { y | ps } ) <-> E. z ( A. t ( t e. z <-> [ t / x ] ph ) /\ [ z / y ] ps ) )
6	1 5	bitri	|- ( { x | ph } e. { y | ps } <-> E. z ( A. t ( t e. z <-> [ t / x ] ph ) /\ [ z / y ] ps ) )