Theorem 4exdistr 1911

Description: Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Assertion

Ref	Expression
4exdistr	|- (E.xE.yE.zE.w((ph /\ ps) /\ (ch /\ th)) <-> E.x(ph /\ E.y(ps /\ E.z(ch /\ E.wth))))

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

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

Proof of Theorem 4exdistr

Step	Hyp	Ref	Expression
1		anass 630	. . . . . . . 8 |- (((ph /\ ps) /\ (ch /\ th)) <-> (ph /\ (ps /\ (ch /\ th))))
2	1	exbii 1582	. . . . . . 7 |- (E.w((ph /\ ps) /\ (ch /\ th)) <-> E.w(ph /\ (ps /\ (ch /\ th))))
3		19.42v 1905	. . . . . . . 8 |- (E.w(ph /\ (ps /\ (ch /\ th))) <-> (ph /\ E.w(ps /\ (ch /\ th))))
4		19.42v 1905	. . . . . . . . 9 |- (E.w(ps /\ (ch /\ th)) <-> (ps /\ E.w(ch /\ th)))
5	4	anbi2i 675	. . . . . . . 8 |- ((ph /\ E.w(ps /\ (ch /\ th))) <-> (ph /\ (ps /\ E.w(ch /\ th))))
6		19.42v 1905	. . . . . . . . . 10 |- (E.w(ch /\ th) <-> (ch /\ E.wth))
7	6	anbi2i 675	. . . . . . . . 9 |- ((ps /\ E.w(ch /\ th)) <-> (ps /\ (ch /\ E.wth)))
8	7	anbi2i 675	. . . . . . . 8 |- ((ph /\ (ps /\ E.w(ch /\ th))) <-> (ph /\ (ps /\ (ch /\ E.wth))))
9	3, 5, 8	3bitri 262	. . . . . . 7 |- (E.w(ph /\ (ps /\ (ch /\ th))) <-> (ph /\ (ps /\ (ch /\ E.wth))))
10	2, 9	bitri 240	. . . . . 6 |- (E.w((ph /\ ps) /\ (ch /\ th)) <-> (ph /\ (ps /\ (ch /\ E.wth))))
11	10	exbii 1582	. . . . 5 |- (E.zE.w((ph /\ ps) /\ (ch /\ th)) <-> E.z(ph /\ (ps /\ (ch /\ E.wth))))
12		19.42v 1905	. . . . 5 |- (E.z(ph /\ (ps /\ (ch /\ E.wth))) <-> (ph /\ E.z(ps /\ (ch /\ E.wth))))
13		19.42v 1905	. . . . . 6 |- (E.z(ps /\ (ch /\ E.wth)) <-> (ps /\ E.z(ch /\ E.wth)))
14	13	anbi2i 675	. . . . 5 |- ((ph /\ E.z(ps /\ (ch /\ E.wth))) <-> (ph /\ (ps /\ E.z(ch /\ E.wth))))
15	11, 12, 14	3bitri 262	. . . 4 |- (E.zE.w((ph /\ ps) /\ (ch /\ th)) <-> (ph /\ (ps /\ E.z(ch /\ E.wth))))
16	15	exbii 1582	. . 3 |- (E.yE.zE.w((ph /\ ps) /\ (ch /\ th)) <-> E.y(ph /\ (ps /\ E.z(ch /\ E.wth))))
17		19.42v 1905	. . 3 |- (E.y(ph /\ (ps /\ E.z(ch /\ E.wth))) <-> (ph /\ E.y(ps /\ E.z(ch /\ E.wth))))
18	16, 17	bitri 240	. 2 |- (E.yE.zE.w((ph /\ ps) /\ (ch /\ th)) <-> (ph /\ E.y(ps /\ E.z(ch /\ E.wth))))
19	18	exbii 1582	1 |- (E.xE.yE.zE.w((ph /\ ps) /\ (ch /\ th)) <-> E.x(ph /\ E.y(ps /\ E.z(ch /\ E.wth))))

Syntax hints:   <-> wb 176   /\ wa 358  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-ex 1542  df-nf 1545

This theorem is referenced by: (None)