Theorem ee4anv

Description: Distribute two pairs of existential quantifiers over a conjunction. For a version requiring fewer axioms but with additional disjoint variable conditions, see 4exdistrv . (Contributed by NM, 31-Jul-1995) Remove disjoint variable conditions on y , z and x , w . (Revised by Eric Schmidt, 26-Oct-2025)

	Ref	Expression
Assertion	ee4anv	|- ( E. x E. y E. z E. w ( ph /\ ps ) <-> ( E. x E. y ph /\ E. z E. w ps ) )

Proof

Step	Hyp	Ref	Expression
1		excom	|- ( E. y E. z E. w ( ph /\ ps ) <-> E. z E. y E. w ( ph /\ ps ) )
2	1	exbii	|- ( E. x E. y E. z E. w ( ph /\ ps ) <-> E. x E. z E. y E. w ( ph /\ ps ) )
3		eeanv	|- ( E. y E. w ( ph /\ ps ) <-> ( E. y ph /\ E. w ps ) )
4	3	2exbii	|- ( E. x E. z E. y E. w ( ph /\ ps ) <-> E. x E. z ( E. y ph /\ E. w ps ) )
5		nfv	|- F/ z ph
6	5	nfex	|- F/ z E. y ph
7		nfv	|- F/ x ps
8	7	nfex	|- F/ x E. w ps
9	6 8	eean	|- ( E. x E. z ( E. y ph /\ E. w ps ) <-> ( E. x E. y ph /\ E. z E. w ps ) )
10	2 4 9	3bitri	|- ( E. x E. y E. z E. w ( ph /\ ps ) <-> ( E. x E. y ph /\ E. z E. w ps ) )