Metamath Proof Explorer


Theorem 19.41vvv

Description: Version of 19.41 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995)

Ref Expression
Assertion 19.41vvv
|- ( E. x E. y E. z ( ph /\ ps ) <-> ( E. x E. y E. z ph /\ ps ) )

Proof

Step Hyp Ref Expression
1 19.41vv
 |-  ( E. y E. z ( ph /\ ps ) <-> ( E. y E. z ph /\ ps ) )
2 1 exbii
 |-  ( E. x E. y E. z ( ph /\ ps ) <-> E. x ( E. y E. z ph /\ ps ) )
3 19.41v
 |-  ( E. x ( E. y E. z ph /\ ps ) <-> ( E. x E. y E. z ph /\ ps ) )
4 2 3 bitri
 |-  ( E. x E. y E. z ( ph /\ ps ) <-> ( E. x E. y E. z ph /\ ps ) )