Theorem 3exdistr 1908

Description: Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
3exdistr	|- ( 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, z

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

Proof of Theorem 3exdistr

Step	Hyp	Ref	Expression
1		3anass 977	. . . 4 |- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) )
2	1	2exbii 1599	. . 3 |- ( E. y E. z ( ph /\ ps /\ ch ) <-> E. y E. z ( ph /\ ( ps /\ ch ) ) )
3		19.42vv 1904	. . 3 |- ( E. y E. z ( ph /\ ( ps /\ ch ) ) <-> ( ph /\ E. y E. z ( ps /\ ch ) ) )
4		exdistr 1902	. . . 4 |- ( E. y E. z ( ps /\ ch ) <-> E. y ( ps /\ E. z ch ) )
5	4	anbi2i 454	. . 3 |- ( ( ph /\ E. y E. z ( ps /\ ch ) ) <-> ( ph /\ E. y ( ps /\ E. z ch ) ) )
6	2, 3, 5	3bitri 205	. 2 |- ( E. y E. z ( ph /\ ps /\ ch ) <-> ( ph /\ E. y ( ps /\ E. z ch ) ) )
7	6	exbii 1598	1 |- ( E. x E. y E. z ( ph /\ ps /\ ch ) <-> E. x ( ph /\ E. y ( ps /\ E. z ch ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 973   E.wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-3an 975

This theorem is referenced by: (None)