Theorem 3exdistr 970

Description: Distribution of existential quantifiers.

Assertion

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

Distinct variable group(s):   ph,y   ph,z   ps,z

Proof of Theorem 3exdistr

Step	Hyp	Ref	Expression
1		3anass 585	. . . . . 6 |- ((ph /\ ps /\ ch) <-> (ph /\ (ps /\ ch)))
2	1	biex 733	. . . . 5 |- (E.z(ph /\ ps /\ ch) <-> E.z(ph /\ (ps /\ ch)))
3		19.42v 966	. . . . 5 |- (E.z(ph /\ (ps /\ ch)) <-> (ph /\ E.z(ps /\ ch)))
4		19.42v 966	. . . . . 6 |- (E.z(ps /\ ch) <-> (ps /\ E.zch))
5	4	anbi2i 367	. . . . 5 |- ((ph /\ E.z(ps /\ ch)) <-> (ph /\ (ps /\ E.zch)))
6	2, 3, 5	3bitr 155	. . . 4 |- (E.z(ph /\ ps /\ ch) <-> (ph /\ (ps /\ E.zch)))
7	6	biex 733	. . 3 |- (E.yE.z(ph /\ ps /\ ch) <-> E.y(ph /\ (ps /\ E.zch)))
8		19.42v 966	. . 3 |- (E.y(ph /\ (ps /\ E.zch)) <-> (ph /\ E.y(ps /\ E.zch)))
9	7, 8	bitr 151	. 2 |- (E.yE.z(ph /\ ps /\ ch) <-> (ph /\ E.y(ps /\ E.zch)))
10	9	biex 733	1 |- (E.xE.yE.z(ph /\ ps /\ ch) <-> E.x(ph /\ E.y(ps /\ E.zch)))

Syntax hints:   <-> wb 127   /\ wa 196   /\ w3a 581  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-3an 583  df-ex 679

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