# Metamath Proof Explorer

## Theorem elelb

Description: Equivalence between two common ways to characterize elements of a class B : the LHS says that sets are elements of B if and only if they satisfy ph while the RHS says that classes are elements of B if and only if they are sets and satisfy ph . Therefore, the LHS is a characterization among sets while the RHS is a characterization among classes. Note that the LHS is often formulated using a class variable instead of the universe _V while this is not possible for the RHS (apart from using B itself, which would not be very useful). (Contributed by BJ, 26-Feb-2023)

Ref Expression
Assertion elelb ${⊢}\left({A}\in \mathrm{V}\to \left({A}\in {B}↔{\phi }\right)\right)↔\left({A}\in {B}↔\left({A}\in \mathrm{V}\wedge {\phi }\right)\right)$

### Proof

Step Hyp Ref Expression
1 elex ${⊢}{A}\in {B}\to {A}\in \mathrm{V}$
2 1 biadani ${⊢}\left({A}\in \mathrm{V}\to \left({A}\in {B}↔{\phi }\right)\right)↔\left({A}\in {B}↔\left({A}\in \mathrm{V}\wedge {\phi }\right)\right)$