Theorem ceqsex6v 2781

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

Hypotheses

Ref	Expression
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

Ref	Expression
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 )

Distinct variable groups:   x, y, z, w, v, u, A   x, B, y, z, w, v, u   x, C, y, z, w, v, u   x, D, y, z, w, v, u   x, E, y, z, w, v, u   x, F, y, z, w, v, u   ps, x   ch, y   th, z   ta, w   et, v   ze, u

Allowed substitution hints:   ph( x, y, z, w, v, u)   ps( y, z, w, v, u)   ch( x, z, w, v, u)   th( x, y, w, v, u)   ta( x, y, z, v, u)   et( x, y, z, w, u)   ze( x, y, z, w, v)

Proof of Theorem ceqsex6v

Step	Hyp	Ref	Expression
1		3anass 982	. . . . 5 |- ( ( ( 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 ) ) )
2	1	3exbii 1607	. . . 4 |- ( 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 ) ) )
3		19.42vvv 1912	. . . 4 |- ( 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 ) ) )
4	2, 3	bitri 184	. . 3 |- ( 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 ) ) )
5	4	3exbii 1607	. 2 |- ( 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 ) ) )
6		ceqsex6v.1	. . . 4 |- A e. _V
7		ceqsex6v.2	. . . 4 |- B e. _V
8		ceqsex6v.3	. . . 4 |- C e. _V
9		ceqsex6v.7	. . . . . 6 |- ( x = A -> ( ph <-> ps ) )
10	9	anbi2d 464	. . . . 5 |- ( x = A -> ( ( ( w = D /\ v = E /\ u = F ) /\ ph ) <-> ( ( w = D /\ v = E /\ u = F ) /\ ps ) ) )
11	10	3exbidv 1869	. . . 4 |- ( 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 ) ) )
12		ceqsex6v.8	. . . . . 6 |- ( y = B -> ( ps <-> ch ) )
13	12	anbi2d 464	. . . . 5 |- ( y = B -> ( ( ( w = D /\ v = E /\ u = F ) /\ ps ) <-> ( ( w = D /\ v = E /\ u = F ) /\ ch ) ) )
14	13	3exbidv 1869	. . . 4 |- ( 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 ) ) )
15		ceqsex6v.9	. . . . . 6 |- ( z = C -> ( ch <-> th ) )
16	15	anbi2d 464	. . . . 5 |- ( z = C -> ( ( ( w = D /\ v = E /\ u = F ) /\ ch ) <-> ( ( w = D /\ v = E /\ u = F ) /\ th ) ) )
17	16	3exbidv 1869	. . . 4 |- ( 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 ) ) )
18	6, 7, 8, 11, 14, 17	ceqsex3v 2779	. . 3 |- ( 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 ) )
19		ceqsex6v.4	. . . 4 |- D e. _V
20		ceqsex6v.5	. . . 4 |- E e. _V
21		ceqsex6v.6	. . . 4 |- F e. _V
22		ceqsex6v.10	. . . 4 |- ( w = D -> ( th <-> ta ) )
23		ceqsex6v.11	. . . 4 |- ( v = E -> ( ta <-> et ) )
24		ceqsex6v.12	. . . 4 |- ( u = F -> ( et <-> ze ) )
25	19, 20, 21, 22, 23, 24	ceqsex3v 2779	. . 3 |- ( E. w E. v E. u ( ( w = D /\ v = E /\ u = F ) /\ th ) <-> ze )
26	18, 25	bitri 184	. 2 |- ( 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 ) ) <-> ze )
27	5, 26	bitri 184	1 |- ( 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 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739

This theorem is referenced by: (None)