Theorem 4exdistr 1931

Description: Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)

Assertion

Ref	Expression
4exdistr	|- ( E. x E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> E. x ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) )

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

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

Proof of Theorem 4exdistr

Step	Hyp	Ref	Expression
1		anass 401	. . . . . . . 8 |- ( ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ph /\ ( ps /\ ( ch /\ th ) ) ) )
2	1	exbii 1619	. . . . . . 7 |- ( E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> E. w ( ph /\ ( ps /\ ( ch /\ th ) ) ) )
3		19.42v 1921	. . . . . . . 8 |- ( E. w ( ph /\ ( ps /\ ( ch /\ th ) ) ) <-> ( ph /\ E. w ( ps /\ ( ch /\ th ) ) ) )
4		19.42v 1921	. . . . . . . . 9 |- ( E. w ( ps /\ ( ch /\ th ) ) <-> ( ps /\ E. w ( ch /\ th ) ) )
5	4	anbi2i 457	. . . . . . . 8 |- ( ( ph /\ E. w ( ps /\ ( ch /\ th ) ) ) <-> ( ph /\ ( ps /\ E. w ( ch /\ th ) ) ) )
6		19.42v 1921	. . . . . . . . . 10 |- ( E. w ( ch /\ th ) <-> ( ch /\ E. w th ) )
7	6	anbi2i 457	. . . . . . . . 9 |- ( ( ps /\ E. w ( ch /\ th ) ) <-> ( ps /\ ( ch /\ E. w th ) ) )
8	7	anbi2i 457	. . . . . . . 8 |- ( ( ph /\ ( ps /\ E. w ( ch /\ th ) ) ) <-> ( ph /\ ( ps /\ ( ch /\ E. w th ) ) ) )
9	3, 5, 8	3bitri 206	. . . . . . 7 |- ( E. w ( ph /\ ( ps /\ ( ch /\ th ) ) ) <-> ( ph /\ ( ps /\ ( ch /\ E. w th ) ) ) )
10	2, 9	bitri 184	. . . . . 6 |- ( E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ph /\ ( ps /\ ( ch /\ E. w th ) ) ) )
11	10	exbii 1619	. . . . 5 |- ( E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> E. z ( ph /\ ( ps /\ ( ch /\ E. w th ) ) ) )
12		19.42v 1921	. . . . 5 |- ( E. z ( ph /\ ( ps /\ ( ch /\ E. w th ) ) ) <-> ( ph /\ E. z ( ps /\ ( ch /\ E. w th ) ) ) )
13		19.42v 1921	. . . . . 6 |- ( E. z ( ps /\ ( ch /\ E. w th ) ) <-> ( ps /\ E. z ( ch /\ E. w th ) ) )
14	13	anbi2i 457	. . . . 5 |- ( ( ph /\ E. z ( ps /\ ( ch /\ E. w th ) ) ) <-> ( ph /\ ( ps /\ E. z ( ch /\ E. w th ) ) ) )
15	11, 12, 14	3bitri 206	. . . 4 |- ( E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ph /\ ( ps /\ E. z ( ch /\ E. w th ) ) ) )
16	15	exbii 1619	. . 3 |- ( E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> E. y ( ph /\ ( ps /\ E. z ( ch /\ E. w th ) ) ) )
17		19.42v 1921	. . 3 |- ( E. y ( ph /\ ( ps /\ E. z ( ch /\ E. w th ) ) ) <-> ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) )
18	16, 17	bitri 184	. 2 |- ( E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) )
19	18	exbii 1619	1 |- ( E. x E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> E. x ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   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-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

This theorem is referenced by: (None)