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Metamath Proof Explorer

Theorem ee4anv

Description: Distribute two pairs of existential quantifiers over a conjunction. For a version requiring fewer axioms but with additional disjoint variable conditions, see 4exdistrv . (Contributed by NM, 31-Jul-1995) Remove disjoint variable conditions on y , z and x , w . (Revised by Eric Schmidt, 26-Oct-2025)

Ref Expression
Assertion ee4anv 
|- ( E. x E. y E. z E. w ( ph /\ ps ) <-> ( E. x E. y ph /\ E. z E. w ps ) )

Proof

Step Hyp Ref Expression
1 excom 
 |-  ( E. y E. z E. w ( ph /\ ps ) <-> E. z E. y E. w ( ph /\ ps ) )
2 1 exbii 
 |-  ( E. x E. y E. z E. w ( ph /\ ps ) <-> E. x E. z E. y E. w ( ph /\ ps ) )
3 eeanv 
 |-  ( E. y E. w ( ph /\ ps ) <-> ( E. y ph /\ E. w ps ) )
4 3 2exbii 
 |-  ( E. x E. z E. y E. w ( ph /\ ps ) <-> E. x E. z ( E. y ph /\ E. w ps ) )
5 nfv 
 |-  F/ z ph
6 5 nfex 
 |-  F/ z E. y ph
7 nfv 
 |-  F/ x ps
8 7 nfex 
 |-  F/ x E. w ps
9 6 8 eean 
 |-  ( E. x E. z ( E. y ph /\ E. w ps ) <-> ( E. x E. y ph /\ E. z E. w ps ) )
10 2 4 9 3bitri 
 |-  ( E. x E. y E. z E. w ( ph /\ ps ) <-> ( E. x E. y ph /\ E. z E. w ps ) )