Theorem 4exdistr 1924

Description: Distribution of existential quantifiers in a quadruple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Wolf Lammen, 20-Jan-2018.)

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		19.42v 1918	. . . . 5 |- ( E. w ( ch /\ th ) <-> ( ch /\ E. w th ) )
2	1	anbi2i 730	. . . 4 |- ( ( ( ph /\ ps ) /\ E. w ( ch /\ th ) ) <-> ( ( ph /\ ps ) /\ ( ch /\ E. w th ) ) )
3		19.42v 1918	. . . 4 |- ( E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ( ph /\ ps ) /\ E. w ( ch /\ th ) ) )
4		df-3an 1039	. . . 4 |- ( ( ph /\ ps /\ ( ch /\ E. w th ) ) <-> ( ( ph /\ ps ) /\ ( ch /\ E. w th ) ) )
5	2, 3, 4	3bitr4i 292	. . 3 |- ( E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> ( ph /\ ps /\ ( ch /\ E. w th ) ) )
6	5	3exbii 1776	. 2 |- ( E. x E. y E. z E. w ( ( ph /\ ps ) /\ ( ch /\ th ) ) <-> E. x E. y E. z ( ph /\ ps /\ ( ch /\ E. w th ) ) )
7		3exdistr 1923	. 2 |- ( E. x E. y E. z ( ph /\ ps /\ ( ch /\ E. w th ) ) <-> E. x ( ph /\ E. y ( ps /\ E. z ( ch /\ E. w th ) ) ) )
8	6, 7	bitri 264	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:   <-> wb 196   /\ wa 384   /\ w3a 1037   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039  df-ex 1705

This theorem is referenced by: (None)