Theorem eeeanv 1914

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 936	. . 3 |- ((ph /\ ps /\ ch) <-> ((ph /\ ps) /\ ch))
2	1	3exbii 1584	. 2 |- (E.xE.yE.z(ph /\ ps /\ ch) <-> E.xE.yE.z((ph /\ ps) /\ ch))
3		eeanv 1913	. . 3 |- (E.yE.z((ph /\ ps) /\ ch) <-> (E.y(ph /\ ps) /\ E.zch))
4	3	exbii 1582	. 2 |- (E.xE.yE.z((ph /\ ps) /\ ch) <-> E.x(E.y(ph /\ ps) /\ E.zch))
5		eeanv 1913	. . . 4 |- (E.xE.y(ph /\ ps) <-> (E.xph /\ E.yps))
6	5	anbi1i 676	. . 3 |- ((E.xE.y(ph /\ ps) /\ E.zch) <-> ((E.xph /\ E.yps) /\ E.zch))
7		19.41v 1901	. . 3 |- (E.x(E.y(ph /\ ps) /\ E.zch) <-> (E.xE.y(ph /\ ps) /\ E.zch))
8		df-3an 936	. . 3 |- ((E.xph /\ E.yps /\ E.zch) <-> ((E.xph /\ E.yps) /\ E.zch))
9	6, 7, 8	3bitr4i 268	. 2 |- (E.x(E.y(ph /\ ps) /\ E.zch) <-> (E.xph /\ E.yps /\ E.zch))
10	2, 4, 9	3bitri 262	1 |- (E.xE.yE.z(ph /\ ps /\ ch) <-> (E.xph /\ E.yps /\ E.zch))

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-ex 1542  df-nf 1545

This theorem is referenced by:  vtocl3  2912  spc3egv  2944  eloprabga  5579  mucass  6136  taddc  6230  letc  6232