Theorem eliminable-abelv

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

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

Proof

Step	Hyp	Ref	Expression
1		dfclel	|- ( { x | ph } e. y <-> E. z ( z = { x | ph } /\ z e. y ) )
2		eliminable-veqab	|- ( z = { x | ph } <-> A. t ( t e. z <-> [ t / x ] ph ) )
3	2	anbi1i	|- ( ( z = { x | ph } /\ z e. y ) <-> ( A. t ( t e. z <-> [ t / x ] ph ) /\ z e. y ) )
4	3	exbii	|- ( E. z ( z = { x | ph } /\ z e. y ) <-> E. z ( A. t ( t e. z <-> [ t / x ] ph ) /\ z e. y ) )
5	1 4	bitri	|- ( { x | ph } e. y <-> E. z ( A. t ( t e. z <-> [ t / x ] ph ) /\ z e. y ) )