Theorem eean

Description: Distribute existential quantifiers. (Contributed by NM, 27-Oct-2010) (Revised by Mario Carneiro, 6-Oct-2016)

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

Proof

Step	Hyp	Ref	Expression
1		eean.1	|- F/ y ph
2		eean.2	|- F/ x ps
3	1	19.42	|- ( E. y ( ph /\ ps ) <-> ( ph /\ E. y ps ) )
4	3	exbii	|- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) )
5	2	nfex	|- F/ x E. y ps
6	5	19.41	|- ( E. x ( ph /\ E. y ps ) <-> ( E. x ph /\ E. y ps ) )
7	4 6	bitri	|- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) )