# Metamath Proof Explorer

## Theorem cleljust

Description: When the class variables in definition df-clel 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 with the class variables in wcel . (Contributed by NM, 28-Jan-2004) Revised to use equsexvw in order to remove dependencies on ax-10 , ax-12 , ax-13 . Note that there is no disjoint variable condition on x , y , that is, on the variables of the left-hand side, as should be the case for definitions. (Revised by BJ, 29-Dec-2020)

Ref Expression
Assertion cleljust ${⊢}{x}\in {y}↔\exists {z}\phantom{\rule{.4em}{0ex}}\left({z}={x}\wedge {z}\in {y}\right)$

### Proof

Step Hyp Ref Expression
1 elequ1 ${⊢}{z}={x}\to \left({z}\in {y}↔{x}\in {y}\right)$
2 1 equsexvw ${⊢}\exists {z}\phantom{\rule{.4em}{0ex}}\left({z}={x}\wedge {z}\in {y}\right)↔{x}\in {y}$
3 2 bicomi ${⊢}{x}\in {y}↔\exists {z}\phantom{\rule{.4em}{0ex}}\left({z}={x}\wedge {z}\in {y}\right)$