Theorem eeeanv 981

Description: Rearrange existential quantifiers.

Assertion

Ref	Expression
eeeanv	|- (E.xE.yE.z(ph /\ ps /\ ch) <-> (E.xph /\ E.yps /\ E.zch))

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

Proof of Theorem eeeanv

Step	Hyp	Ref	Expression
1		19.42vv 968	. . . . 5 |- (E.yE.z(ph /\ (ps /\ ch)) <-> (ph /\ E.yE.z(ps /\ ch)))
2		eeanv 980	. . . . . 6 |- (E.yE.z(ps /\ ch) <-> (E.yps /\ E.zch))
3	2	anbi2i 367	. . . . 5 |- ((ph /\ E.yE.z(ps /\ ch)) <-> (ph /\ (E.yps /\ E.zch)))
4	1, 3	bitr 151	. . . 4 |- (E.yE.z(ph /\ (ps /\ ch)) <-> (ph /\ (E.yps /\ E.zch)))
5	4	biex 733	. . 3 |- (E.xE.yE.z(ph /\ (ps /\ ch)) <-> E.x(ph /\ (E.yps /\ E.zch)))
6		19.41v 963	. . 3 |- (E.x(ph /\ (E.yps /\ E.zch)) <-> (E.xph /\ (E.yps /\ E.zch)))
7	5, 6	bitr 151	. 2 |- (E.xE.yE.z(ph /\ (ps /\ ch)) <-> (E.xph /\ (E.yps /\ E.zch)))
8		3anass 585	. . 3 |- ((ph /\ ps /\ ch) <-> (ph /\ (ps /\ ch)))
9	8	bi3ex 735	. 2 |- (E.xE.yE.z(ph /\ ps /\ ch) <-> E.xE.yE.z(ph /\ (ps /\ ch)))
10		3anass 585	. 2 |- ((E.xph /\ E.yps /\ E.zch) <-> (E.xph /\ (E.yps /\ E.zch)))
11	7, 9, 10	3bitr4 158	1 |- (E.xE.yE.z(ph /\ ps /\ ch) <-> (E.xph /\ E.yps /\ E.zch))

Syntax hints:   <-> wb 127   /\ wa 196   /\ w3a 581  E.wex 678

This theorem is referenced by:  vtocl3 1380  eloprabg 3035

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-7 676  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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