Theorem eeeanv 1326

Description: Rearrange existential quantifiers.

Assertion

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

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

Proof of Theorem eeeanv

Step	Hyp	Ref	Expression
1		19.42vv 1312	. . . . 5 |- (E.yE.z(ph /\ (ps /\ ch)) <-> (ph /\ E.yE.z(ps /\ ch)))
2		eeanv 1325	. . . . . 6 |- (E.yE.z(ps /\ ch) <-> (E.yps /\ E.zch))
3	2	anbi2i 482	. . . . 5 |- ((ph /\ E.yE.z(ps /\ ch)) <-> (ph /\ (E.yps /\ E.zch)))
4	1, 3	bitr 173	. . . 4 |- (E.yE.z(ph /\ (ps /\ ch)) <-> (ph /\ (E.yps /\ E.zch)))
5	4	exbii 1053	. . 3 |- (E.xE.yE.z(ph /\ (ps /\ ch)) <-> E.x(ph /\ (E.yps /\ E.zch)))
6		19.41v 1307	. . 3 |- (E.x(ph /\ (E.yps /\ E.zch)) <-> (E.xph /\ (E.yps /\ E.zch)))
7	5, 6	bitr 173	. 2 |- (E.xE.yE.z(ph /\ (ps /\ ch)) <-> (E.xph /\ (E.yps /\ E.zch)))
8		3anass 781	. . 3 |- ((ph /\ ps /\ ch) <-> (ph /\ (ps /\ ch)))
9	8	3exbi 1055	. 2 |- (E.xE.yE.z(ph /\ ps /\ ch) <-> E.xE.yE.z(ph /\ (ps /\ ch)))
10		3anass 781	. 2 |- ((E.xph /\ E.yps /\ E.zch) <-> (E.xph /\ (E.yps /\ E.zch)))
11	7, 9, 10	3bitr4 183	1 |- (E.xE.yE.z(ph /\ ps /\ ch) <-> (E.xph /\ E.yps /\ E.zch))

Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 777  E.wex 982

This theorem is referenced by:  vtocl3 1847  cla43egv 1869  eloprabg 4013  eeeeanv 10431

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983

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