Metamath Proof Explorer

Theorem ceqsex3v

Description: Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011)

Ref Expression
Hypotheses ceqsex3v.1 
|- A e. _V
ceqsex3v.2 
|- B e. _V
ceqsex3v.3 
|- C e. _V
ceqsex3v.4 
|- ( x = A -> ( ph <-> ps ) )
ceqsex3v.5 
|- ( y = B -> ( ps <-> ch ) )
ceqsex3v.6 
|- ( z = C -> ( ch <-> th ) )
Assertion ceqsex3v 
|- ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> th )

Proof

Step Hyp Ref Expression
1 ceqsex3v.1 
 |-  A e. _V
2 ceqsex3v.2 
 |-  B e. _V
3 ceqsex3v.3 
 |-  C e. _V
4 ceqsex3v.4 
 |-  ( x = A -> ( ph <-> ps ) )
5 ceqsex3v.5 
 |-  ( y = B -> ( ps <-> ch ) )
6 ceqsex3v.6 
 |-  ( z = C -> ( ch <-> th ) )
7 anass 
 |-  ( ( ( x = A /\ ( y = B /\ z = C ) ) /\ ph ) <-> ( x = A /\ ( ( y = B /\ z = C ) /\ ph ) ) )
8 3anass 
 |-  ( ( x = A /\ y = B /\ z = C ) <-> ( x = A /\ ( y = B /\ z = C ) ) )
9 8 anbi1i 
 |-  ( ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> ( ( x = A /\ ( y = B /\ z = C ) ) /\ ph ) )
10 df-3an 
 |-  ( ( y = B /\ z = C /\ ph ) <-> ( ( y = B /\ z = C ) /\ ph ) )
11 10 anbi2i 
 |-  ( ( x = A /\ ( y = B /\ z = C /\ ph ) ) <-> ( x = A /\ ( ( y = B /\ z = C ) /\ ph ) ) )
12 7 9 11 3bitr4i 
 |-  ( ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> ( x = A /\ ( y = B /\ z = C /\ ph ) ) )
13 12 2exbii 
 |-  ( E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> E. y E. z ( x = A /\ ( y = B /\ z = C /\ ph ) ) )
14 19.42vv 
 |-  ( E. y E. z ( x = A /\ ( y = B /\ z = C /\ ph ) ) <-> ( x = A /\ E. y E. z ( y = B /\ z = C /\ ph ) ) )
15 13 14 bitri 
 |-  ( E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> ( x = A /\ E. y E. z ( y = B /\ z = C /\ ph ) ) )
16 15 exbii 
 |-  ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> E. x ( x = A /\ E. y E. z ( y = B /\ z = C /\ ph ) ) )
17 4 3anbi3d 
 |-  ( x = A -> ( ( y = B /\ z = C /\ ph ) <-> ( y = B /\ z = C /\ ps ) ) )
18 17 2exbidv 
 |-  ( x = A -> ( E. y E. z ( y = B /\ z = C /\ ph ) <-> E. y E. z ( y = B /\ z = C /\ ps ) ) )
19 1 18 ceqsexv 
 |-  ( E. x ( x = A /\ E. y E. z ( y = B /\ z = C /\ ph ) ) <-> E. y E. z ( y = B /\ z = C /\ ps ) )
20 2 3 5 6 ceqsex2v 
 |-  ( E. y E. z ( y = B /\ z = C /\ ps ) <-> th )
21 19 20 bitri 
 |-  ( E. x ( x = A /\ E. y E. z ( y = B /\ z = C /\ ph ) ) <-> th )
22 16 21 bitri 
 |-  ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> th )