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Metamath Proof Explorer

Theorem ceqsex6v

Description: Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011)

Ref Expression
Hypotheses ceqsex6v.1 
|- A e. _V
ceqsex6v.2 
|- B e. _V
ceqsex6v.3 
|- C e. _V
ceqsex6v.4 
|- D e. _V
ceqsex6v.5 
|- E e. _V
ceqsex6v.6 
|- F e. _V
ceqsex6v.7 
|- ( x = A -> ( ph <-> ps ) )
ceqsex6v.8 
|- ( y = B -> ( ps <-> ch ) )
ceqsex6v.9 
|- ( z = C -> ( ch <-> th ) )
ceqsex6v.10 
|- ( w = D -> ( th <-> ta ) )
ceqsex6v.11 
|- ( v = E -> ( ta <-> et ) )
ceqsex6v.12 
|- ( u = F -> ( et <-> ze ) )
Assertion ceqsex6v 
|- ( E. x E. y E. z E. w E. v E. u ( ( x = A /\ y = B /\ z = C ) /\ ( w = D /\ v = E /\ u = F ) /\ ph ) <-> ze )

Proof

Step Hyp Ref Expression
1 ceqsex6v.1 
 |-  A e. _V
2 ceqsex6v.2 
 |-  B e. _V
3 ceqsex6v.3 
 |-  C e. _V
4 ceqsex6v.4 
 |-  D e. _V
5 ceqsex6v.5 
 |-  E e. _V
6 ceqsex6v.6 
 |-  F e. _V
7 ceqsex6v.7 
 |-  ( x = A -> ( ph <-> ps ) )
8 ceqsex6v.8 
 |-  ( y = B -> ( ps <-> ch ) )
9 ceqsex6v.9 
 |-  ( z = C -> ( ch <-> th ) )
10 ceqsex6v.10 
 |-  ( w = D -> ( th <-> ta ) )
11 ceqsex6v.11 
 |-  ( v = E -> ( ta <-> et ) )
12 ceqsex6v.12 
 |-  ( u = F -> ( et <-> ze ) )
13 3anass 
 |-  ( ( ( x = A /\ y = B /\ z = C ) /\ ( w = D /\ v = E /\ u = F ) /\ ph ) <-> ( ( x = A /\ y = B /\ z = C ) /\ ( ( w = D /\ v = E /\ u = F ) /\ ph ) ) )
14 13 3exbii 
 |-  ( E. w E. v E. u ( ( x = A /\ y = B /\ z = C ) /\ ( w = D /\ v = E /\ u = F ) /\ ph ) <-> E. w E. v E. u ( ( x = A /\ y = B /\ z = C ) /\ ( ( w = D /\ v = E /\ u = F ) /\ ph ) ) )
15 19.42vvv 
 |-  ( E. w E. v E. u ( ( x = A /\ y = B /\ z = C ) /\ ( ( w = D /\ v = E /\ u = F ) /\ ph ) ) <-> ( ( x = A /\ y = B /\ z = C ) /\ E. w E. v E. u ( ( w = D /\ v = E /\ u = F ) /\ ph ) ) )
16 14 15 bitri 
 |-  ( E. w E. v E. u ( ( x = A /\ y = B /\ z = C ) /\ ( w = D /\ v = E /\ u = F ) /\ ph ) <-> ( ( x = A /\ y = B /\ z = C ) /\ E. w E. v E. u ( ( w = D /\ v = E /\ u = F ) /\ ph ) ) )
17 16 3exbii 
 |-  ( E. x E. y E. z E. w E. v E. u ( ( x = A /\ y = B /\ z = C ) /\ ( w = D /\ v = E /\ u = F ) /\ ph ) <-> E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ E. w E. v E. u ( ( w = D /\ v = E /\ u = F ) /\ ph ) ) )
18 7 anbi2d 
 |-  ( x = A -> ( ( ( w = D /\ v = E /\ u = F ) /\ ph ) <-> ( ( w = D /\ v = E /\ u = F ) /\ ps ) ) )
19 18 3exbidv 
 |-  ( x = A -> ( E. w E. v E. u ( ( w = D /\ v = E /\ u = F ) /\ ph ) <-> E. w E. v E. u ( ( w = D /\ v = E /\ u = F ) /\ ps ) ) )
20 8 anbi2d 
 |-  ( y = B -> ( ( ( w = D /\ v = E /\ u = F ) /\ ps ) <-> ( ( w = D /\ v = E /\ u = F ) /\ ch ) ) )
21 20 3exbidv 
 |-  ( y = B -> ( E. w E. v E. u ( ( w = D /\ v = E /\ u = F ) /\ ps ) <-> E. w E. v E. u ( ( w = D /\ v = E /\ u = F ) /\ ch ) ) )
22 9 anbi2d 
 |-  ( z = C -> ( ( ( w = D /\ v = E /\ u = F ) /\ ch ) <-> ( ( w = D /\ v = E /\ u = F ) /\ th ) ) )
23 22 3exbidv 
 |-  ( z = C -> ( E. w E. v E. u ( ( w = D /\ v = E /\ u = F ) /\ ch ) <-> E. w E. v E. u ( ( w = D /\ v = E /\ u = F ) /\ th ) ) )
24 1 2 3 19 21 23 ceqsex3v 
 |-  ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ E. w E. v E. u ( ( w = D /\ v = E /\ u = F ) /\ ph ) ) <-> E. w E. v E. u ( ( w = D /\ v = E /\ u = F ) /\ th ) )
25 4 5 6 10 11 12 ceqsex3v 
 |-  ( E. w E. v E. u ( ( w = D /\ v = E /\ u = F ) /\ th ) <-> ze )
26 17 24 25 3bitri 
 |-  ( E. x E. y E. z E. w E. v E. u ( ( x = A /\ y = B /\ z = C ) /\ ( w = D /\ v = E /\ u = F ) /\ ph ) <-> ze )