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Metamath Proof Explorer

Theorem 4exdistr

Description: Distribution of existential quantifiers in a quadruple conjunction. (Contributed by NM, 9-Mar-1995) (Proof shortened by Wolf Lammen, 20-Jan-2018)

Ref Expression
Assertion 4exdistr 
|- ( E. x E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> E. x ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) )

Proof

Step Hyp Ref Expression
1 19.42v 
 |-  ( E. w ( ch /\ th ) <-> ( ch /\ E. w th ) )
2 1 anbi2i 
 |-  ( ( ( ph /\ ps ) /\ E. w ( ch /\ th ) ) <-> ( ( ph /\ ps ) /\ ( ch /\ E. w th ) ) )
3 19.42v 
 |-  ( E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ( ph /\ ps ) /\ E. w ( ch /\ th ) ) )
4 df-3an 
 |-  ( ( ph /\ ps /\ ( ch /\ E. w th ) ) <-> ( ( ph /\ ps ) /\ ( ch /\ E. w th ) ) )
5 2 3 4 3bitr4i 
 |-  ( E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ph /\ ps /\ ( ch /\ E. w th ) ) )
6 5 3exbii 
 |-  ( E. x E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> E. x E. y E. z ( ph /\ ps /\ ( ch /\ E. w th ) ) )
7 3exdistr 
 |-  ( E. x E. y E. z ( ph /\ ps /\ ( ch /\ E. w th ) ) <-> E. x ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) )
8 6 7 bitri 
 |-  ( E. x E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> E. x ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) )