Theorem ee4anv 1915

Description: Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)

Assertion

Ref	Expression
ee4anv	|- (E.xE.yE.zE.w(ph /\ ps) <-> (E.xE.yph /\ E.zE.wps))

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

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

Proof of Theorem ee4anv

Step	Hyp	Ref	Expression
1		excom 1741	. . 3 |- (E.yE.zE.w(ph /\ ps) <-> E.zE.yE.w(ph /\ ps))
2	1	exbii 1582	. 2 |- (E.xE.yE.zE.w(ph /\ ps) <-> E.xE.zE.yE.w(ph /\ ps))
3		eeanv 1913	. . 3 |- (E.yE.w(ph /\ ps) <-> (E.yph /\ E.wps))
4	3	2exbii 1583	. 2 |- (E.xE.zE.yE.w(ph /\ ps) <-> E.xE.z(E.yph /\ E.wps))
5		eeanv 1913	. 2 |- (E.xE.z(E.yph /\ E.wps) <-> (E.xE.yph /\ E.zE.wps))
6	2, 4, 5	3bitri 262	1 |- (E.xE.yE.zE.w(ph /\ ps) <-> (E.xE.yph /\ E.zE.wps))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  cgsex4g  2892  funsi  5520  fntxp  5804  fnpprod  5843