Theorem ceqsex3v 2898

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.xE.yE.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 630	. . . . . 6 |- (((x = A /\ (y = B /\ z = C)) /\ ph) <-> (x = A /\ ((y = B /\ z = C) /\ ph)))
2		3anass 938	. . . . . . 7 |- ((x = A /\ y = B /\ z = C) <-> (x = A /\ (y = B /\ z = C)))
3	2	anbi1i 676	. . . . . 6 |- (((x = A /\ y = B /\ z = C) /\ ph) <-> ((x = A /\ (y = B /\ z = C)) /\ ph))
4		df-3an 936	. . . . . . 7 |- ((y = B /\ z = C /\ ph) <-> ((y = B /\ z = C) /\ ph))
5	4	anbi2i 675	. . . . . 6 |- ((x = A /\ (y = B /\ z = C /\ ph)) <-> (x = A /\ ((y = B /\ z = C) /\ ph)))
6	1, 3, 5	3bitr4i 268	. . . . 5 |- (((x = A /\ y = B /\ z = C) /\ ph) <-> (x = A /\ (y = B /\ z = C /\ ph)))
7	6	2exbii 1583	. . . 4 |- (E.yE.z((x = A /\ y = B /\ z = C) /\ ph) <-> E.yE.z(x = A /\ (y = B /\ z = C /\ ph)))
8		19.42vv 1907	. . . 4 |- (E.yE.z(x = A /\ (y = B /\ z = C /\ ph)) <-> (x = A /\ E.yE.z(y = B /\ z = C /\ ph)))
9	7, 8	bitri 240	. . 3 |- (E.yE.z((x = A /\ y = B /\ z = C) /\ ph) <-> (x = A /\ E.yE.z(y = B /\ z = C /\ ph)))
10	9	exbii 1582	. 2 |- (E.xE.yE.z((x = A /\ y = B /\ z = C) /\ ph) <-> E.x(x = A /\ E.yE.z(y = B /\ z = C /\ ph)))
11		ceqsex3v.1	. . . 4 |- A e. _V
12		ceqsex3v.4	. . . . . 6 |- (x = A -> (ph <-> ps))
13	12	3anbi3d 1258	. . . . 5 |- (x = A -> ((y = B /\ z = C /\ ph) <-> (y = B /\ z = C /\ ps)))
14	13	2exbidv 1628	. . . 4 |- (x = A -> (E.yE.z(y = B /\ z = C /\ ph) <-> E.yE.z(y = B /\ z = C /\ ps)))
15	11, 14	ceqsexv 2895	. . 3 |- (E.x(x = A /\ E.yE.z(y = B /\ z = C /\ ph)) <-> E.yE.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 2897	. . 3 |- (E.yE.z(y = B /\ z = C /\ ps) <-> th)
21	15, 20	bitri 240	. 2 |- (E.x(x = A /\ E.yE.z(y = B /\ z = C /\ ph)) <-> th)
22	10, 21	bitri 240	1 |- (E.xE.yE.z((x = A /\ y = B /\ z = C) /\ ph) <-> th)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  ceqsex6v  2900