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Metamath Proof Explorer

Theorem raltpg

Description: Convert a restricted universal quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011) (Revised by Mario Carneiro, 23-Apr-2015)

Ref Expression
Hypotheses ralprg.1 
|- ( x = A -> ( ph <-> ps ) )
ralprg.2 
|- ( x = B -> ( ph <-> ch ) )
raltpg.3 
|- ( x = C -> ( ph <-> th ) )
Assertion raltpg 
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) ) )

Proof

Step Hyp Ref Expression
1 ralprg.1 
 |-  ( x = A -> ( ph <-> ps ) )
2 ralprg.2 
 |-  ( x = B -> ( ph <-> ch ) )
3 raltpg.3 
 |-  ( x = C -> ( ph <-> th ) )
4 1 2 ralprg 
 |-  ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) )
5 3 ralsng 
 |-  ( C e. X -> ( A. x e. { C } ph <-> th ) )
6 4 5 bi2anan9 
 |-  ( ( ( A e. V /\ B e. W ) /\ C e. X ) -> ( ( A. x e. { A , B } ph /\ A. x e. { C } ph ) <-> ( ( ps /\ ch ) /\ th ) ) )
7 6 3impa 
 |-  ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A. x e. { A , B } ph /\ A. x e. { C } ph ) <-> ( ( ps /\ ch ) /\ th ) ) )
8 df-tp 
 |-  { A , B , C } = ( { A , B } u. { C } )
9 8 raleqi 
 |-  ( A. x e. { A , B , C } ph <-> A. x e. ( { A , B } u. { C } ) ph )
10 ralunb 
 |-  ( A. x e. ( { A , B } u. { C } ) ph <-> ( A. x e. { A , B } ph /\ A. x e. { C } ph ) )
11 9 10 bitri 
 |-  ( A. x e. { A , B , C } ph <-> ( A. x e. { A , B } ph /\ A. x e. { C } ph ) )
12 df-3an 
 |-  ( ( ps /\ ch /\ th ) <-> ( ( ps /\ ch ) /\ th ) )
13 7 11 12 3bitr4g 
 |-  ( ( A e. V /\ B e. W /\ C e. X ) -> ( A. x e. { A , B , C } ph <-> ( ps /\ ch /\ th ) ) )