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Metamath Proof Explorer

Theorem 19.41vvvv

Description: Version of 19.41 with four quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by FL, 14-Jul-2007)

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

Proof

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