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Metamath Proof Explorer

Theorem elab6g

Description: Membership in a class abstraction. Class version of sb6 . (Contributed by SN, 5-Oct-2024)

Ref Expression
Assertion elab6g 
|- ( A e. V -> ( A e. { x | ph } <-> A. x ( x = A -> ph ) ) )

Proof

Step Hyp Ref Expression
1 eleq1 
 |-  ( y = A -> ( y e. { x | ph } <-> A e. { x | ph } ) )
2 eqeq2 
 |-  ( y = A -> ( x = y <-> x = A ) )
3 2 imbi1d 
 |-  ( y = A -> ( ( x = y -> ph ) <-> ( x = A -> ph ) ) )
4 3 albidv 
 |-  ( y = A -> ( A. x ( x = y -> ph ) <-> A. x ( x = A -> ph ) ) )
5 df-clab 
 |-  ( y e. { x | ph } <-> [ y / x ] ph )
6 sb6 
 |-  ( [ y / x ] ph <-> A. x ( x = y -> ph ) )
7 5 6 bitri 
 |-  ( y e. { x | ph } <-> A. x ( x = y -> ph ) )
8 1 4 7 vtoclbg 
 |-  ( A e. V -> ( A e. { x | ph } <-> A. x ( x = A -> ph ) ) )