Theorem ceqsex4v 2769

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. x E. y E. z E. 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 1899	. . . 4 |- ( E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D /\ ph ) ) <-> ( ( x = A /\ y = B ) /\ E. z E. w ( z = C /\ w = D /\ ph ) ) )
2		3anass 972	. . . . . 6 |- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ ph ) <-> ( ( x = A /\ y = B ) /\ ( ( z = C /\ w = D ) /\ ph ) ) )
3		df-3an 970	. . . . . . 7 |- ( ( z = C /\ w = D /\ ph ) <-> ( ( z = C /\ w = D ) /\ ph ) )
4	3	anbi2i 453	. . . . . 6 |- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D /\ ph ) ) <-> ( ( x = A /\ y = B ) /\ ( ( z = C /\ w = D ) /\ ph ) ) )
5	2, 4	bitr4i 186	. . . . 5 |- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ ph ) <-> ( ( x = A /\ y = B ) /\ ( z = C /\ w = D /\ ph ) ) )
6	5	2exbii 1594	. . . 4 |- ( E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ ph ) <-> E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D /\ ph ) ) )
7		df-3an 970	. . . 4 |- ( ( x = A /\ y = B /\ E. z E. w ( z = C /\ w = D /\ ph ) ) <-> ( ( x = A /\ y = B ) /\ E. z E. w ( z = C /\ w = D /\ ph ) ) )
8	1, 6, 7	3bitr4i 211	. . 3 |- ( E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ ph ) <-> ( x = A /\ y = B /\ E. z E. w ( z = C /\ w = D /\ ph ) ) )
9	8	2exbii 1594	. 2 |- ( E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ ph ) <-> E. x E. y ( x = A /\ y = B /\ E. z E. 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 1308	. . . 4 |- ( x = A -> ( ( z = C /\ w = D /\ ph ) <-> ( z = C /\ w = D /\ ps ) ) )
14	13	2exbidv 1856	. . 3 |- ( x = A -> ( E. z E. w ( z = C /\ w = D /\ ph ) <-> E. z E. w ( z = C /\ w = D /\ ps ) ) )
15		ceqsex4v.8	. . . . 5 |- ( y = B -> ( ps <-> ch ) )
16	15	3anbi3d 1308	. . . 4 |- ( y = B -> ( ( z = C /\ w = D /\ ps ) <-> ( z = C /\ w = D /\ ch ) ) )
17	16	2exbidv 1856	. . 3 |- ( y = B -> ( E. z E. w ( z = C /\ w = D /\ ps ) <-> E. z E. w ( z = C /\ w = D /\ ch ) ) )
18	10, 11, 14, 17	ceqsex2v 2767	. 2 |- ( E. x E. y ( x = A /\ y = B /\ E. z E. w ( z = C /\ w = D /\ ph ) ) <-> E. z E. 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 2767	. 2 |- ( E. z E. w ( z = C /\ w = D /\ ch ) <-> ta )
24	9, 18, 23	3bitri 205	1 |- ( E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ ph ) <-> ta )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by:  ceqsex8v  2771