Theorem eeeanv 2044

Description: Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) Reduce distinct variable restrictions. (Revised by Wolf Lammen, 20-Jan-2018.)

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   ps, x   ps, z   ch, x   ch, y

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

Proof of Theorem eeeanv

Step	Hyp	Ref	Expression
1		eeanv 2043	. . 3 |- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) )
2	1	anbi1i 699	. 2 |- ( ( E. x E. y ( ph /\ ps ) /\ E. z ch ) <-> ( ( E. x ph /\ E. y ps ) /\ E. z ch ) )
3		df-3an 984	. . . . . 6 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
4	3	exbii 1712	. . . . 5 |- ( E. z ( ph /\ ps /\ ch ) <-> E. z ( ( ph /\ ps ) /\ ch ) )
5		19.42v 1823	. . . . 5 |- ( E. z ( ( ph /\ ps ) /\ ch ) <-> ( ( ph /\ ps ) /\ E. z ch ) )
6	4, 5	bitri 252	. . . 4 |- ( E. z ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ E. z ch ) )
7	6	2exbii 1713	. . 3 |- ( E. x E. y E. z ( ph /\ ps /\ ch ) <-> E. x E. y ( ( ph /\ ps ) /\ E. z ch ) )
8		nfv 1751	. . . . . 6 |- F/ y ch
9	8	nfex 2004	. . . . 5 |- F/ y E. z ch
10	9	19.41 2026	. . . 4 |- ( E. y ( ( ph /\ ps ) /\ E. z ch ) <-> ( E. y ( ph /\ ps ) /\ E. z ch ) )
11	10	exbii 1712	. . 3 |- ( E. x E. y ( ( ph /\ ps ) /\ E. z ch ) <-> E. x ( E. y ( ph /\ ps ) /\ E. z ch ) )
12		nfv 1751	. . . . 5 |- F/ x ch
13	12	nfex 2004	. . . 4 |- F/ x E. z ch
14	13	19.41 2026	. . 3 |- ( E. x ( E. y ( ph /\ ps ) /\ E. z ch ) <-> ( E. x E. y ( ph /\ ps ) /\ E. z ch ) )
15	7, 11, 14	3bitri 274	. 2 |- ( E. x E. y E. z ( ph /\ ps /\ ch ) <-> ( E. x E. y ( ph /\ ps ) /\ E. z ch ) )
16		df-3an 984	. 2 |- ( ( E. x ph /\ E. y ps /\ E. z ch ) <-> ( ( E. x ph /\ E. y ps ) /\ E. z ch ) )
17	2, 15, 16	3bitr4i 280	1 |- ( E. x E. y E. z ( ph /\ ps /\ ch ) <-> ( E. x ph /\ E. y ps /\ E. z ch ) )

Syntax hints:   <-> wb 187   /\ wa 370   /\ w3a 982   E.wex 1659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905

This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-ex 1660  df-nf 1664

This theorem is referenced by:  vtocl3  3135  spc3egv  3170  eloprabga  6394