Theorem ee4anv 982

Description: Rearrange existential quantifiers.

Assertion

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

Distinct variable group(s):   ph,z   ph,w   ps,x   ps,y   y,z   x,w

Proof of Theorem ee4anv

Step	Hyp	Ref	Expression
1		excom 728	. . 3 |- (E.yE.zE.w(ph /\ ps) <-> E.zE.yE.w(ph /\ ps))
2	1	biex 733	. 2 |- (E.xE.yE.zE.w(ph /\ ps) <-> E.xE.zE.yE.w(ph /\ ps))
3		eeanv 980	. . 3 |- (E.yE.w(ph /\ ps) <-> (E.yph /\ E.wps))
4	3	bi2ex 734	. 2 |- (E.xE.zE.yE.w(ph /\ ps) <-> E.xE.z(E.yph /\ E.wps))
5		eeanv 980	. 2 |- (E.xE.z(E.yph /\ E.wps) <-> (E.xE.yph /\ E.zE.wps))
6	2, 4, 5	3bitr 155	1 |- (E.xE.yE.zE.w(ph /\ ps) <-> (E.xE.yph /\ E.zE.wps))

Syntax hints:   <-> wb 127   /\ wa 196  E.wex 678

This theorem is referenced by:  cgsex4g 1369  th3qlem1 3250  distrlem5pr 3925  5oalem7 5550  3oalem3 5554

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679

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