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Metamath Proof Explorer

Theorem clel4g

Description: Alternate definition of membership in a set. (Contributed by NM, 18-Aug-1993) Strengthen from sethood hypothesis to sethood antecedent and avoid ax-12 . (Revised by BJ, 1-Sep-2024)

Ref Expression
Assertion clel4g 
|- ( B e. V -> ( A e. B <-> A. x ( x = B -> A e. x ) ) )

Proof

Step Hyp Ref Expression
1 elisset 
 |-  ( B e. V -> E. x x = B )
2 biimt 
 |-  ( E. x x = B -> ( A e. B <-> ( E. x x = B -> A e. B ) ) )
3 1 2 syl 
 |-  ( B e. V -> ( A e. B <-> ( E. x x = B -> A e. B ) ) )
4 19.23v 
 |-  ( A. x ( x = B -> A e. B ) <-> ( E. x x = B -> A e. B ) )
5 3 4 bitr4di 
 |-  ( B e. V -> ( A e. B <-> A. x ( x = B -> A e. B ) ) )
6 eleq2 
 |-  ( x = B -> ( A e. x <-> A e. B ) )
7 6 bicomd 
 |-  ( x = B -> ( A e. B <-> A e. x ) )
8 7 pm5.74i 
 |-  ( ( x = B -> A e. B ) <-> ( x = B -> A e. x ) )
9 8 albii 
 |-  ( A. x ( x = B -> A e. B ) <-> A. x ( x = B -> A e. x ) )
10 5 9 bitrdi 
 |-  ( B e. V -> ( A e. B <-> A. x ( x = B -> A e. x ) ) )