Theorem 4exdistr 971

Description: Distribution of existential quantifiers.

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 group(s):   ph,y   ph,z   ph,w   ps,z   ps,w   ch,w

Proof of Theorem 4exdistr

Step	Hyp	Ref	Expression
1		anass 336	. . . . . . . 8 |- (((ph /\ ps) /\ (ch /\ th)) <-> (ph /\ (ps /\ (ch /\ th))))
2	1	biex 733	. . . . . . 7 |- (E.w((ph /\ ps) /\ (ch /\ th)) <-> E.w(ph /\ (ps /\ (ch /\ th))))
3		19.42v 966	. . . . . . . 8 |- (E.w(ph /\ (ps /\ (ch /\ th))) <-> (ph /\ E.w(ps /\ (ch /\ th))))
4		19.42v 966	. . . . . . . . 9 |- (E.w(ps /\ (ch /\ th)) <-> (ps /\ E.w(ch /\ th)))
5	4	anbi2i 367	. . . . . . . 8 |- ((ph /\ E.w(ps /\ (ch /\ th))) <-> (ph /\ (ps /\ E.w(ch /\ th))))
6		19.42v 966	. . . . . . . . . 10 |- (E.w(ch /\ th) <-> (ch /\ E.wth))
7	6	anbi2i 367	. . . . . . . . 9 |- ((ps /\ E.w(ch /\ th)) <-> (ps /\ (ch /\ E.wth)))
8	7	anbi2i 367	. . . . . . . 8 |- ((ph /\ (ps /\ E.w(ch /\ th))) <-> (ph /\ (ps /\ (ch /\ E.wth))))
9	3, 5, 8	3bitr 155	. . . . . . 7 |- (E.w(ph /\ (ps /\ (ch /\ th))) <-> (ph /\ (ps /\ (ch /\ E.wth))))
10	2, 9	bitr 151	. . . . . 6 |- (E.w((ph /\ ps) /\ (ch /\ th)) <-> (ph /\ (ps /\ (ch /\ E.wth))))
11	10	biex 733	. . . . 5 |- (E.zE.w((ph /\ ps) /\ (ch /\ th)) <-> E.z(ph /\ (ps /\ (ch /\ E.wth))))
12		19.42v 966	. . . . 5 |- (E.z(ph /\ (ps /\ (ch /\ E.wth))) <-> (ph /\ E.z(ps /\ (ch /\ E.wth))))
13		19.42v 966	. . . . . 6 |- (E.z(ps /\ (ch /\ E.wth)) <-> (ps /\ E.z(ch /\ E.wth)))
14	13	anbi2i 367	. . . . 5 |- ((ph /\ E.z(ps /\ (ch /\ E.wth))) <-> (ph /\ (ps /\ E.z(ch /\ E.wth))))
15	11, 12, 14	3bitr 155	. . . 4 |- (E.zE.w((ph /\ ps) /\ (ch /\ th)) <-> (ph /\ (ps /\ E.z(ch /\ E.wth))))
16	15	biex 733	. . 3 |- (E.yE.zE.w((ph /\ ps) /\ (ch /\ th)) <-> E.y(ph /\ (ps /\ E.z(ch /\ E.wth))))
17		19.42v 966	. . 3 |- (E.y(ph /\ (ps /\ E.z(ch /\ E.wth))) <-> (ph /\ E.y(ps /\ E.z(ch /\ E.wth))))
18	16, 17	bitr 151	. 2 |- (E.yE.zE.w((ph /\ ps) /\ (ch /\ th)) <-> (ph /\ E.y(ps /\ E.z(ch /\ E.wth))))
19	18	biex 733	1 |- (E.xE.yE.zE.w((ph /\ ps) /\ (ch /\ th)) <-> E.x(ph /\ E.y(ps /\ E.z(ch /\ E.wth))))

Syntax hints:   <-> wb 127   /\ wa 196  E.wex 678

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679

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