Theorem 4exdistr

Description: Distribution of existential quantifiers in a quadruple conjunction. (Contributed by NM, 9-Mar-1995) (Proof shortened by Wolf Lammen, 20-Jan-2018)

	Ref	Expression
Assertion	4exdistr	|- ( E. x E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> E. x ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		19.42v	|- ( E. w ( ch /\ th ) <-> ( ch /\ E. w th ) )
2	1	anbi2i	|- ( ( ( ph /\ ps ) /\ E. w ( ch /\ th ) ) <-> ( ( ph /\ ps ) /\ ( ch /\ E. w th ) ) )
3		19.42v	|- ( E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ( ph /\ ps ) /\ E. w ( ch /\ th ) ) )
4		df-3an	|- ( ( ph /\ ps /\ ( ch /\ E. w th ) ) <-> ( ( ph /\ ps ) /\ ( ch /\ E. w th ) ) )
5	2 3 4	3bitr4i	|- ( E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ph /\ ps /\ ( ch /\ E. w th ) ) )
6	5	3exbii	|- ( E. x E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> E. x E. y E. z ( ph /\ ps /\ ( ch /\ E. w th ) ) )
7		3exdistr	|- ( E. x E. y E. z ( ph /\ ps /\ ( ch /\ E. w th ) ) <-> E. x ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) )
8	6 7	bitri	|- ( E. x E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> E. x ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) )