Theorem reuanid

Description: Cancellation law for restricted unique existential quantification. (Contributed by Peter Mazsa, 12-Feb-2018)

	Ref	Expression
Assertion	reuanid	|- ( E! x e. A ( x e. A /\ ph ) <-> E! x e. A ph )

Proof

Step	Hyp	Ref	Expression
1		anabs5	|- ( ( x e. A /\ ( x e. A /\ ph ) ) <-> ( x e. A /\ ph ) )
2	1	eubii	|- ( E! x ( x e. A /\ ( x e. A /\ ph ) ) <-> E! x ( x e. A /\ ph ) )
3		df-reu	|- ( E! x e. A ( x e. A /\ ph ) <-> E! x ( x e. A /\ ( x e. A /\ ph ) ) )
4		df-reu	|- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
5	2 3 4	3bitr4i	|- ( E! x e. A ( x e. A /\ ph ) <-> E! x e. A ph )