Theorem cleljust 1810

Description: When the class variables of set theory are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 1391 with the class variables in wcel 1390. (Contributed by NM, 28-Jan-2004.)

Assertion

Ref	Expression
cleljust	|- (x e. y <-> E.z(z = x /\ z e. y))

Distinct variable groups:   x,z   y,z

Proof of Theorem cleljust

Step	Hyp	Ref	Expression
1		ax-17 1416	. . 3 |- (x e. y -> A.z x e. y)
2		elequ1 1597	. . 3 |- (z = x -> (z e. y <-> x e. y))
3	1, 2	equsex 1613	. 2 |- (E.z(z = x /\ z e. y) <-> x e. y)
4	3	bicomi 123	1 |- (x e. y <-> E.z(z = x /\ z e. y))

Syntax hints:   /\ wa 97   <-> wb 98  E.wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-13 1401  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)