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 )