Theorem cleljust 1912

Description: When the class variables in definition df-clel 2467 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 1905 with the class variables in wcel 1904. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 1856 in order to remove dependencies on ax-10 1932, ax-12 1950, ax-13 2104. Note that there is no DV condition on x , y, that is, on the variables of the left-hand side. (Revised by BJ, 29-Dec-2020.)

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		elequ1 1911	. . 3 |- ( z = x -> ( z e. y <-> x e. y ) )
2	1	equsexvw 1856	. 2 |- ( E. z ( z = x /\ z e. y ) <-> x e. y )
3	2	bicomi 207	1 |- ( x e. y <-> E. z ( z = x /\ z e. y ) )

Syntax hints:   <-> wb 189   /\ wa 376   E.wex 1671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906

This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672

This theorem is referenced by:  bj-dfclel  31565