Theorem ceqsex4v 2591

Description: Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)

Hypotheses

Ref	Expression
ceqsex4v.1	|- A e. _V
ceqsex4v.2	|- B e. _V
ceqsex4v.3	|- C e. _V
ceqsex4v.4	|- D e. _V
ceqsex4v.7	|- (x = A -> (ph <-> ps))
ceqsex4v.8	|- (y = B -> (ps <-> ch))
ceqsex4v.9	|- (z = C -> (ch <-> th))
ceqsex4v.10	|- (w = D -> (th <-> ta))

Assertion

Ref	Expression
ceqsex4v	|- (E.xE.yE.zE.w((x = A /\ y = B) /\ (z = C /\ w = D) /\ ph) <-> ta)

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

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

Proof of Theorem ceqsex4v

Step	Hyp	Ref	Expression
1		19.42vv 1785	. . . 4 |- (E.zE.w((x = A /\ y = B) /\ (z = C /\ w = D /\ ph)) <-> ((x = A /\ y = B) /\ E.zE.w(z = C /\ w = D /\ ph)))
2		3anass 888	. . . . . 6 |- (((x = A /\ y = B) /\ (z = C /\ w = D) /\ ph) <-> ((x = A /\ y = B) /\ ((z = C /\ w = D) /\ ph)))
3		df-3an 886	. . . . . . 7 |- ((z = C /\ w = D /\ ph) <-> ((z = C /\ w = D) /\ ph))
4	3	anbi2i 430	. . . . . 6 |- (((x = A /\ y = B) /\ (z = C /\ w = D /\ ph)) <-> ((x = A /\ y = B) /\ ((z = C /\ w = D) /\ ph)))
5	2, 4	bitr4i 176	. . . . 5 |- (((x = A /\ y = B) /\ (z = C /\ w = D) /\ ph) <-> ((x = A /\ y = B) /\ (z = C /\ w = D /\ ph)))
6	5	2exbii 1494	. . . 4 |- (E.zE.w((x = A /\ y = B) /\ (z = C /\ w = D) /\ ph) <-> E.zE.w((x = A /\ y = B) /\ (z = C /\ w = D /\ ph)))
7		df-3an 886	. . . 4 |- ((x = A /\ y = B /\ E.zE.w(z = C /\ w = D /\ ph)) <-> ((x = A /\ y = B) /\ E.zE.w(z = C /\ w = D /\ ph)))
8	1, 6, 7	3bitr4i 201	. . 3 |- (E.zE.w((x = A /\ y = B) /\ (z = C /\ w = D) /\ ph) <-> (x = A /\ y = B /\ E.zE.w(z = C /\ w = D /\ ph)))
9	8	2exbii 1494	. 2 |- (E.xE.yE.zE.w((x = A /\ y = B) /\ (z = C /\ w = D) /\ ph) <-> E.xE.y(x = A /\ y = B /\ E.zE.w(z = C /\ w = D /\ ph)))
10		ceqsex4v.1	. . 3 |- A e. _V
11		ceqsex4v.2	. . 3 |- B e. _V
12		ceqsex4v.7	. . . . 5 |- (x = A -> (ph <-> ps))
13	12	3anbi3d 1212	. . . 4 |- (x = A -> ((z = C /\ w = D /\ ph) <-> (z = C /\ w = D /\ ps)))
14	13	2exbidv 1745	. . 3 |- (x = A -> (E.zE.w(z = C /\ w = D /\ ph) <-> E.zE.w(z = C /\ w = D /\ ps)))
15		ceqsex4v.8	. . . . 5 |- (y = B -> (ps <-> ch))
16	15	3anbi3d 1212	. . . 4 |- (y = B -> ((z = C /\ w = D /\ ps) <-> (z = C /\ w = D /\ ch)))
17	16	2exbidv 1745	. . 3 |- (y = B -> (E.zE.w(z = C /\ w = D /\ ps) <-> E.zE.w(z = C /\ w = D /\ ch)))
18	10, 11, 14, 17	ceqsex2v 2589	. 2 |- (E.xE.y(x = A /\ y = B /\ E.zE.w(z = C /\ w = D /\ ph)) <-> E.zE.w(z = C /\ w = D /\ ch))
19		ceqsex4v.3	. . 3 |- C e. _V
20		ceqsex4v.4	. . 3 |- D e. _V
21		ceqsex4v.9	. . 3 |- (z = C -> (ch <-> th))
22		ceqsex4v.10	. . 3 |- (w = D -> (th <-> ta))
23	19, 20, 21, 22	ceqsex2v 2589	. 2 |- (E.zE.w(z = C /\ w = D /\ ch) <-> ta)
24	9, 18, 23	3bitri 195	1 |- (E.xE.yE.zE.w((x = A /\ y = B) /\ (z = C /\ w = D) /\ ph) <-> ta)

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:  ceqsex8v  2593