Theorem 3exdistr 1927

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 984	. . . 4 |- ( ( ph /\ ps /\ ch ) <-> ( ph /\ ( ps /\ ch ) ) )
2	1	2exbii 1617	. . 3 |- ( E. y E. z ( ph /\ ps /\ ch ) <-> E. y E. z ( ph /\ ( ps /\ ch ) ) )
3		19.42vv 1923	. . 3 |- ( E. y E. z ( ph /\ ( ps /\ ch ) ) <-> ( ph /\ E. y E. z ( ps /\ ch ) ) )
4		exdistr 1921	. . . 4 |- ( E. y E. z ( ps /\ ch ) <-> E. y ( ps /\ E. z ch ) )
5	4	anbi2i 457	. . 3 |- ( ( ph /\ E. y E. z ( ps /\ ch ) ) <-> ( ph /\ E. y ( ps /\ E. z ch ) ) )
6	2, 3, 5	3bitri 206	. 2 |- ( E. y E. z ( ph /\ ps /\ ch ) <-> ( ph /\ E. y ( ps /\ E. z ch ) ) )
7	6	exbii 1616	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 1503

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545

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

This theorem is referenced by: (None)