Theorem reean

Description: Rearrange restricted existential quantifiers. For a version based on fewer axioms see reeanv . (Contributed by NM, 27-Oct-2010) (Proof shortened by Andrew Salmon, 30-May-2011)

	Ref	Expression
Hypotheses	reean.1	|- F/ y ph
	reean.2	|- F/ x ps
Assertion	reean	|- ( E. x e. A E. y e. B ( ph /\ ps ) <-> ( E. x e. A ph /\ E. y e. B ps ) )

Proof

Step	Hyp	Ref	Expression
1		reean.1	|- F/ y ph
2		reean.2	|- F/ x ps
3		nfv	|- F/ y x e. A
4	3 1	nfan	|- F/ y ( x e. A /\ ph )
5		nfv	|- F/ x y e. B
6	5 2	nfan	|- F/ x ( y e. B /\ ps )
7	4 6	eean	|- ( E. x E. y ( ( x e. A /\ ph ) /\ ( y e. B /\ ps ) ) <-> ( E. x ( x e. A /\ ph ) /\ E. y ( y e. B /\ ps ) ) )
8	7	reeanlem	|- ( E. x e. A E. y e. B ( ph /\ ps ) <-> ( E. x e. A ph /\ E. y e. B ps ) )