Theorem 3exdistr

Description: Distribution of existential quantifiers in a triple conjunction. (Contributed by NM, 9-Mar-1995) (Proof shortened by Andrew Salmon, 25-May-2011)

	Ref	Expression
Assertion	3exdistr	|- ( E. x E. y E. z ( ph /\ ps /\ ch ) <-> E. x ( ph /\ E. y ( ps /\ E. z ch ) ) )

Proof

Step	Hyp	Ref	Expression
1		3anass	|- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) )
2	1	2exbii	|- ( E. y E. z ( ph /\ ps /\ ch ) <-> E. y E. z ( ph /\ ( ps /\ ch ) ) )
3		19.42vv	|- ( E. y E. z ( ph /\ ( ps /\ ch ) ) <-> ( ph /\ E. y E. z ( ps /\ ch ) ) )
4		exdistr	|- ( E. y E. z ( ps /\ ch ) <-> E. y ( ps /\ E. z ch ) )
5	4	anbi2i	|- ( ( ph /\ E. y E. z ( ps /\ ch ) ) <-> ( ph /\ E. y ( ps /\ E. z ch ) ) )
6	2 3 5	3bitri	|- ( E. y E. z ( ph /\ ps /\ ch ) <-> ( ph /\ E. y ( ps /\ E. z ch ) ) )
7	6	exbii	|- ( E. x E. y E. z ( ph /\ ps /\ ch ) <-> E. x ( ph /\ E. y ( ps /\ E. z ch ) ) )