Theorem eeeanv 1805

Description: Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
eeeanv	|- (E.xE.yE.z(ph /\ ps /\ ch) <-> (E.xph /\ E.yps /\ E.zch))

Distinct variable groups:   ph,y   ph,z   x,z,ps   x,y,ch

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

Proof of Theorem eeeanv

Step	Hyp	Ref	Expression
1		df-3an 886	. . 3 |- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
2	1	3exbii 1495	. 2 |- (E.xE.yE.z(ph /\ ps /\ ch) <-> E.xE.yE.z((ph /\ ps) /\ ch))
3		eeanv 1804	. . 3 |- (E.yE.z((ph /\ ps) /\ ch) <-> (E.y(ph /\ ps) /\ E.zch))
4	3	exbii 1493	. 2 |- (E.xE.yE.z((ph /\ ps) /\ ch) <-> E.x(E.y(ph /\ ps) /\ E.zch))
5		eeanv 1804	. . . 4 |- (E.xE.y(ph /\ ps) <-> (E.xph /\ E.yps))
6	5	anbi1i 431	. . 3 |- ((E.xE.y(ph /\ ps) /\ E.zch) <-> ((E.xph /\ E.yps) /\ E.zch))
7		19.41v 1779	. . 3 |- (E.x(E.y(ph /\ ps) /\ E.zch) <-> (E.xE.y(ph /\ ps) /\ E.zch))
8		df-3an 886	. . 3 |- ((E.xph /\ E.yps /\ E.zch) <-> ((E.xph /\ E.yps) /\ E.zch))
9	6, 7, 8	3bitr4i 201	. 2 |- (E.x(E.y(ph /\ ps) /\ E.zch) <-> (E.xph /\ E.yps /\ E.zch))
10	2, 4, 9	3bitri 195	1 |- (E.xE.yE.z(ph /\ ps /\ ch) <-> (E.xph /\ E.yps /\ E.zch))

Syntax hints:   /\ wa 97   <-> wb 98   /\ w3a 884  E.wex 1378

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-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347

This theorem is referenced by:  vtocl3  2604  spc3egv  2638  spc3gv  2639  eloprabga  5533  prarloc  6486