Theorem eeeanv 1952

Description: Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
eeeanv	|- ( E. x E. y E. z ( ph /\ ps /\ ch ) <-> ( E. x ph /\ E. y ps /\ E. z ch ) )

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

Allowed substitution hints:   ph( x)   ps( y)   ch( z)

Proof of Theorem eeeanv

Step	Hyp	Ref	Expression
1		df-3an 982	. . 3 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
2	1	3exbii 1621	. 2 |- ( E. x E. y E. z ( ph /\ ps /\ ch ) <-> E. x E. y E. z ( ( ph /\ ps ) /\ ch ) )
3		eeanv 1951	. . 3 |- ( E. y E. z ( ( ph /\ ps ) /\ ch ) <-> ( E. y ( ph /\ ps ) /\ E. z ch ) )
4	3	exbii 1619	. 2 |- ( E. x E. y E. z ( ( ph /\ ps ) /\ ch ) <-> E. x ( E. y ( ph /\ ps ) /\ E. z ch ) )
5		eeanv 1951	. . . 4 |- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) )
6	5	anbi1i 458	. . 3 |- ( ( E. x E. y ( ph /\ ps ) /\ E. z ch ) <-> ( ( E. x ph /\ E. y ps ) /\ E. z ch ) )
7		19.41v 1917	. . 3 |- ( E. x ( E. y ( ph /\ ps ) /\ E. z ch ) <-> ( E. x E. y ( ph /\ ps ) /\ E. z ch ) )
8		df-3an 982	. . 3 |- ( ( E. x ph /\ E. y ps /\ E. z ch ) <-> ( ( E. x ph /\ E. y ps ) /\ E. z ch ) )
9	6, 7, 8	3bitr4i 212	. 2 |- ( E. x ( E. y ( ph /\ ps ) /\ E. z ch ) <-> ( E. x ph /\ E. y ps /\ E. z ch ) )
10	2, 4, 9	3bitri 206	1 |- ( E. x E. y E. z ( ph /\ ps /\ ch ) <-> ( E. x ph /\ E. y ps /\ E. z ch ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 980   E.wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1475

This theorem is referenced by:  vtocl3  2820  spc3egv  2856  spc3gv  2857  eloprabga  6009  prarloc  7570