Theorem 3exdistr 1910

Description: Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
3exdistr	|- (E.xE.yE.z(ph /\ ps /\ ch) <-> E.x(ph /\ E.y(ps /\ E.zch)))

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

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

Proof of Theorem 3exdistr

Step	Hyp	Ref	Expression
1		3anass 938	. . . 4 |- ((ph /\ ps /\ ch) <-> (ph /\ (ps /\ ch)))
2	1	2exbii 1583	. . 3 |- (E.yE.z(ph /\ ps /\ ch) <-> E.yE.z(ph /\ (ps /\ ch)))
3		19.42vv 1907	. . 3 |- (E.yE.z(ph /\ (ps /\ ch)) <-> (ph /\ E.yE.z(ps /\ ch)))
4		exdistr 1906	. . . 4 |- (E.yE.z(ps /\ ch) <-> E.y(ps /\ E.zch))
5	4	anbi2i 675	. . 3 |- ((ph /\ E.yE.z(ps /\ ch)) <-> (ph /\ E.y(ps /\ E.zch)))
6	2, 3, 5	3bitri 262	. 2 |- (E.yE.z(ph /\ ps /\ ch) <-> (ph /\ E.y(ps /\ E.zch)))
7	6	exbii 1582	1 |- (E.xE.yE.z(ph /\ ps /\ ch) <-> E.x(ph /\ E.y(ps /\ 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-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: (None)