Theorem ceqsex3v 2590

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 381	. . . . . 6 |- (((x = A /\ (y = B /\ z = C)) /\ ph) <-> (x = A /\ ((y = B /\ z = C) /\ ph)))
2		3anass 888	. . . . . . 7 |- ((x = A /\ y = B /\ z = C) <-> (x = A /\ (y = B /\ z = C)))
3	2	anbi1i 431	. . . . . 6 |- (((x = A /\ y = B /\ z = C) /\ ph) <-> ((x = A /\ (y = B /\ z = C)) /\ ph))
4		df-3an 886	. . . . . . 7 |- ((y = B /\ z = C /\ ph) <-> ((y = B /\ z = C) /\ ph))
5	4	anbi2i 430	. . . . . 6 |- ((x = A /\ (y = B /\ z = C /\ ph)) <-> (x = A /\ ((y = B /\ z = C) /\ ph)))
6	1, 3, 5	3bitr4i 201	. . . . 5 |- (((x = A /\ y = B /\ z = C) /\ ph) <-> (x = A /\ (y = B /\ z = C /\ ph)))
7	6	2exbii 1494	. . . 4 |- (E.yE.z((x = A /\ y = B /\ z = C) /\ ph) <-> E.yE.z(x = A /\ (y = B /\ z = C /\ ph)))
8		19.42vv 1785	. . . 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 173	. . 3 |- (E.yE.z((x = A /\ y = B /\ z = C) /\ ph) <-> (x = A /\ E.yE.z(y = B /\ z = C /\ ph)))
10	9	exbii 1493	. 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 1212	. . . . 5 |- (x = A -> ((y = B /\ z = C /\ ph) <-> (y = B /\ z = C /\ ps)))
14	13	2exbidv 1745	. . . 4 |- (x = A -> (E.yE.z(y = B /\ z = C /\ ph) <-> E.yE.z(y = B /\ z = C /\ ps)))
15	11, 14	ceqsexv 2587	. . 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 2589	. . 3 |- (E.yE.z(y = B /\ z = C /\ ps) <-> th)
21	15, 20	bitri 173	. 2 |- (E.x(x = A /\ E.yE.z(y = B /\ z = C /\ ph)) <-> th)
22	10, 21	bitri 173	1 |- (E.xE.yE.z((x = A /\ y = B /\ z = C) /\ ph) <-> th)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 884   = wceq 1242  E.wex 1378   e. wcel 1390   _Vcvv 2551

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  ceqsex6v  2592