Theorem r19.41v

Description: Restricted quantifier version 19.41v . Version of r19.41 with a disjoint variable condition, requiring fewer axioms. (Contributed by NM, 17-Dec-2003) Reduce dependencies on axioms. (Revised by BJ, 29-Mar-2020)

	Ref	Expression
Assertion	r19.41v	|- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) )

Proof

Step	Hyp	Ref	Expression
1		anass	|- ( ( ( x e. A /\ ph ) /\ ps ) <-> ( x e. A /\ ( ph /\ ps ) ) )
2	1	exbii	|- ( E. x ( ( x e. A /\ ph ) /\ ps ) <-> E. x ( x e. A /\ ( ph /\ ps ) ) )
3		19.41v	|- ( E. x ( ( x e. A /\ ph ) /\ ps ) <-> ( E. x ( x e. A /\ ph ) /\ ps ) )
4	2 3	bitr3i	|- ( E. x ( x e. A /\ ( ph /\ ps ) ) <-> ( E. x ( x e. A /\ ph ) /\ ps ) )
5		df-rex	|- ( E. x e. A ( ph /\ ps ) <-> E. x ( x e. A /\ ( ph /\ ps ) ) )
6		df-rex	|- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
7	6	anbi1i	|- ( ( E. x e. A ph /\ ps ) <-> ( E. x ( x e. A /\ ph ) /\ ps ) )
8	4 5 7	3bitr4i	|- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) )