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Metamath Proof Explorer

Theorem ralprg

Description: Convert a restricted universal quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011) (Revised by Mario Carneiro, 23-Apr-2015) Avoid ax-10 , ax-12 . (Revised by GG, 30-Sep-2024)

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

Proof

Step Hyp Ref Expression
1 ralprg.1 
 |-  ( x = A -> ( ph <-> ps ) )
2 ralprg.2 
 |-  ( x = B -> ( ph <-> ch ) )
3 df-pr 
 |-  { A , B } = ( { A } u. { B } )
4 3 raleqi 
 |-  ( A. x e. { A , B } ph <-> A. x e. ( { A } u. { B } ) ph )
5 ralunb 
 |-  ( A. x e. ( { A } u. { B } ) ph <-> ( A. x e. { A } ph /\ A. x e. { B } ph ) )
6 4 5 bitri 
 |-  ( A. x e. { A , B } ph <-> ( A. x e. { A } ph /\ A. x e. { B } ph ) )
7 1 ralsng 
 |-  ( A e. V -> ( A. x e. { A } ph <-> ps ) )
8 2 ralsng 
 |-  ( B e. W -> ( A. x e. { B } ph <-> ch ) )
9 7 8 bi2anan9 
 |-  ( ( A e. V /\ B e. W ) -> ( ( A. x e. { A } ph /\ A. x e. { B } ph ) <-> ( ps /\ ch ) ) )
10 6 9 bitrid 
 |-  ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) )