Theorem ceqsex3v 3012

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

Hypotheses

Ref	Expression
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

Ref	Expression
ceqsex3v	|- ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> th )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   ps, x   ch, y   th, z

Allowed substitution hints:   ph( x, y, z)   ps( y, z)   ch( x, z)   th( x, y)

Proof of Theorem ceqsex3v

Step	Hyp	Ref	Expression
1		anass 649	. . . . . 6 |- ( ( ( x = A /\ ( y = B /\ z = C ) ) /\ ph ) <-> ( x = A /\ ( ( y = B /\ z = C ) /\ ph ) ) )
2		3anass 969	. . . . . . 7 |- ( ( x = A /\ y = B /\ z = C ) <-> ( x = A /\ ( y = B /\ z = C ) ) )
3	2	anbi1i 695	. . . . . 6 |- ( ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> ( ( x = A /\ ( y = B /\ z = C ) ) /\ ph ) )
4		df-3an 967	. . . . . . 7 |- ( ( y = B /\ z = C /\ ph ) <-> ( ( y = B /\ z = C ) /\ ph ) )
5	4	anbi2i 694	. . . . . 6 |- ( ( x = A /\ ( y = B /\ z = C /\ ph ) ) <-> ( x = A /\ ( ( y = B /\ z = C ) /\ ph ) ) )
6	1, 3, 5	3bitr4i 277	. . . . 5 |- ( ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> ( x = A /\ ( y = B /\ z = C /\ ph ) ) )
7	6	2exbii 1635	. . . 4 |- ( E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> E. y E. z ( x = A /\ ( y = B /\ z = C /\ ph ) ) )
8		19.42vv 1926	. . . 4 |- ( E. y E. z ( x = A /\ ( y = B /\ z = C /\ ph ) ) <-> ( x = A /\ E. y E. z ( y = B /\ z = C /\ ph ) ) )
9	7, 8	bitri 249	. . 3 |- ( E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> ( x = A /\ E. y E. z ( y = B /\ z = C /\ ph ) ) )
10	9	exbii 1634	. 2 |- ( 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 ) ) )
11		ceqsex3v.1	. . . 4 |- A e. _V
12		ceqsex3v.4	. . . . . 6 |- ( x = A -> ( ph <-> ps ) )
13	12	3anbi3d 1295	. . . . 5 |- ( x = A -> ( ( y = B /\ z = C /\ ph ) <-> ( y = B /\ z = C /\ ps ) ) )
14	13	2exbidv 1682	. . . 4 |- ( x = A -> ( E. y E. z ( y = B /\ z = C /\ ph ) <-> E. y E. z ( y = B /\ z = C /\ ps ) ) )
15	11, 14	ceqsexv 3009	. . 3 |- ( E. x ( x = A /\ E. y E. z ( y = B /\ z = C /\ ph ) ) <-> E. y E. z ( y = B /\ z = C /\ ps ) )
16		ceqsex3v.2	. . . 4 |- B e. _V
17		ceqsex3v.3	. . . 4 |- C e. _V
18		ceqsex3v.5	. . . 4 |- ( y = B -> ( ps <-> ch ) )
19		ceqsex3v.6	. . . 4 |- ( z = C -> ( ch <-> th ) )
20	16, 17, 18, 19	ceqsex2v 3011	. . 3 |- ( E. y E. z ( y = B /\ z = C /\ ps ) <-> th )
21	15, 20	bitri 249	. 2 |- ( E. x ( x = A /\ E. y E. z ( y = B /\ z = C /\ ph ) ) <-> th )
22	10, 21	bitri 249	1 |- ( E. x E. y E. z ( ( x = A /\ y = B /\ z = C ) /\ ph ) <-> th )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   E.wex 1586   e. wcel 1756   _Vcvv 2972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-v 2974

This theorem is referenced by:  ceqsex6v  3014