Theorem cleljust 2082

Description: When the class variables in definition df-clel 2462 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 1768 with the class variables in wcel 1767. Note: This proof is referenced on the Metamath Proof Explorer Home Page and shouldn't be changed. (Contributed by NM, 28-Jan-2004.) (Proof modification is discouraged.)

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-5 1680	. . 3 |- ( x e. y -> A. z x e. y )
2		elequ1 1770	. . 3 |- ( z = x -> ( z e. y <-> x e. y ) )
3	1, 2	equsexh 2012	. 2 |- ( E. z ( z = x /\ z e. y ) <-> x e. y )
4	3	bicomi 202	1 |- ( x e. y <-> E. z ( z = x /\ z e. y ) )

Syntax hints:   <-> wb 184   /\ wa 369   E.wex 1596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-10 1786  ax-12 1803  ax-13 1968

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600

This theorem is referenced by: (None)