Theorem 3exdistr 1310

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 groups:   ph,y   ph,z   ps,z

Proof of Theorem 3exdistr

Step	Hyp	Ref	Expression
1		3anass 778	. . . . . 6 |- ((ph /\ ps /\ ch) <-> (ph /\ (ps /\ ch)))
2	1	exbii 1049	. . . . 5 |- (E.z(ph /\ ps /\ ch) <-> E.z(ph /\ (ps /\ ch)))
3		19.42v 1306	. . . . 5 |- (E.z(ph /\ (ps /\ ch)) <-> (ph /\ E.z(ps /\ ch)))
4		19.42v 1306	. . . . . 6 |- (E.z(ps /\ ch) <-> (ps /\ E.zch))
5	4	anbi2i 480	. . . . 5 |- ((ph /\ E.z(ps /\ ch)) <-> (ph /\ (ps /\ E.zch)))
6	2, 3, 5	3bitr 177	. . . 4 |- (E.z(ph /\ ps /\ ch) <-> (ph /\ (ps /\ E.zch)))
7	6	exbii 1049	. . 3 |- (E.yE.z(ph /\ ps /\ ch) <-> E.y(ph /\ (ps /\ E.zch)))
8		19.42v 1306	. . 3 |- (E.y(ph /\ (ps /\ E.zch)) <-> (ph /\ E.y(ps /\ E.zch)))
9	7, 8	bitr 173	. 2 |- (E.yE.z(ph /\ ps /\ ch) <-> (ph /\ E.y(ps /\ E.zch)))
10	9	exbii 1049	1 |- (E.xE.yE.z(ph /\ ps /\ ch) <-> E.x(ph /\ E.y(ps /\ E.zch)))

Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 774  E.wex 978

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979

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