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Metamath Proof Explorer

Theorem clel2g

Description: Alternate definition of membership when the member is a set. (Contributed by NM, 18-Aug-1993) Strengthen from sethood hypothesis to sethood antecedent. (Revised by BJ, 12-Feb-2022) Avoid ax-12 . (Revised by BJ, 1-Sep-2024)

Ref Expression
Assertion clel2g 
|- ( A e. V -> ( A e. B <-> A. x ( x = A -> x e. B ) ) )

Proof

Step Hyp Ref Expression
1 elisset 
 |-  ( A e. V -> E. x x = A )
2 biimt 
 |-  ( E. x x = A -> ( A e. B <-> ( E. x x = A -> A e. B ) ) )
3 1 2 syl 
 |-  ( A e. V -> ( A e. B <-> ( E. x x = A -> A e. B ) ) )
4 19.23v 
 |-  ( A. x ( x = A -> A e. B ) <-> ( E. x x = A -> A e. B ) )
5 eleq1 
 |-  ( x = A -> ( x e. B <-> A e. B ) )
6 5 bicomd 
 |-  ( x = A -> ( A e. B <-> x e. B ) )
7 6 pm5.74i 
 |-  ( ( x = A -> A e. B ) <-> ( x = A -> x e. B ) )
8 7 albii 
 |-  ( A. x ( x = A -> A e. B ) <-> A. x ( x = A -> x e. B ) )
9 4 8 bitr3i 
 |-  ( ( E. x x = A -> A e. B ) <-> A. x ( x = A -> x e. B ) )
10 3 9 bitrdi 
 |-  ( A e. V -> ( A e. B <-> A. x ( x = A -> x e. B ) ) )