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Metamath Proof Explorer

Theorem rspcimdv

Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017)

Ref Expression
Hypotheses rspcimdv.1 
|- ( ph -> A e. B )
rspcimdv.2 
|- ( ( ph /\ x = A ) -> ( ps -> ch ) )
Assertion rspcimdv 
|- ( ph -> ( A. x e. B ps -> ch ) )

Proof

Step Hyp Ref Expression
1 rspcimdv.1 
 |-  ( ph -> A e. B )
2 rspcimdv.2 
 |-  ( ( ph /\ x = A ) -> ( ps -> ch ) )
3 df-ral 
 |-  ( A. x e. B ps <-> A. x ( x e. B -> ps ) )
4 simpr 
 |-  ( ( ph /\ x = A ) -> x = A )
5 4 eleq1d 
 |-  ( ( ph /\ x = A ) -> ( x e. B <-> A e. B ) )
6 5 biimprd 
 |-  ( ( ph /\ x = A ) -> ( A e. B -> x e. B ) )
7 6 2 imim12d 
 |-  ( ( ph /\ x = A ) -> ( ( x e. B -> ps ) -> ( A e. B -> ch ) ) )
8 1 7 spcimdv 
 |-  ( ph -> ( A. x ( x e. B -> ps ) -> ( A e. B -> ch ) ) )
9 1 8 mpid 
 |-  ( ph -> ( A. x ( x e. B -> ps ) -> ch ) )
10 3 9 biimtrid 
 |-  ( ph -> ( A. x e. B ps -> ch ) )