Theorem ee4anv 1323

Description: Rearrange existential quantifiers.

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

Proof of Theorem ee4anv

Step	Hyp	Ref	Expression
1		excom 1044	. . 3 |- (E.yE.zE.w(ph /\ ps) <-> E.zE.yE.w(ph /\ ps))
2	1	exbii 1049	. 2 |- (E.xE.yE.zE.w(ph /\ ps) <-> E.xE.zE.yE.w(ph /\ ps))
3		eeanv 1321	. . 3 |- (E.yE.w(ph /\ ps) <-> (E.yph /\ E.wps))
4	3	2exbii 1050	. 2 |- (E.xE.zE.yE.w(ph /\ ps) <-> E.xE.z(E.yph /\ E.wps))
5		eeanv 1321	. 2 |- (E.xE.z(E.yph /\ E.wps) <-> (E.xE.yph /\ E.zE.wps))
6	2, 4, 5	3bitr 177	1 |- (E.xE.yE.zE.w(ph /\ ps) <-> (E.xE.yph /\ E.zE.wps))

Syntax hints:   <-> wb 146   /\ wa 223  E.wex 978

This theorem is referenced by:  cgsex4g 1829  th3qlem1 4304  distrlem5pr 5111  5oalem7 9545  3oalem3 9549

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979

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