Theorem cleljust 2182

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 2177 with the class variables in wcel 2176. (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 1549	. . 3 |- ( x e. y -> A. z x e. y )
2		elequ1 2180	. . 3 |- ( z = x -> ( z e. y <-> x e. y ) )
3	1, 2	equsex 1751	. 2 |- ( E. z ( z = x /\ z e. y ) <-> x e. y )
4	3	bicomi 132	1 |- ( x e. y <-> E. z ( z = x /\ z e. y ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-13 2178

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)