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Metamath Proof Explorer

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 ) ) )