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Metamath Proof Explorer

Theorem 19.42vvv

Description: Version of 19.42 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011) (Proof shortened by Wolf Lammen, 27-Aug-2023)

		Ref	Expression
	Assertion	19.42vvv	\|- ( E. x E. y E. z ( ph /\ ps ) <-> ( ph /\ E. x E. y E. z ps ) )

Proof

Step	Hyp	Ref	Expression
1		exdistr2	\|- ( E. x E. y E. z ( ph /\ ps ) <-> E. x ( ph /\ E. y E. z ps ) )
2		19.42v	\|- ( E. x ( ph /\ E. y E. z ps ) <-> ( ph /\ E. x E. y E. z ps ) )
3	1 2	bitri	\|- ( E. x E. y E. z ( ph /\ ps ) <-> ( ph /\ E. x E. y E. z ps ) )