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Metamath Proof Explorer

Theorem ralprgf

Description: Convert a restricted universal quantification over a pair to a conjunction, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 17-Sep-2011) (Revised by AV, 8-Apr-2023)

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

Proof

Step Hyp Ref Expression
1 ralprgf.1 
 |-  F/ x ps
2 ralprgf.2 
 |-  F/ x ch
3 ralprgf.a 
 |-  ( x = A -> ( ph <-> ps ) )
4 ralprgf.b 
 |-  ( x = B -> ( ph <-> ch ) )
5 df-pr 
 |-  { A , B } = ( { A } u. { B } )
6 5 raleqi 
 |-  ( A. x e. { A , B } ph <-> A. x e. ( { A } u. { B } ) ph )
7 ralunb 
 |-  ( A. x e. ( { A } u. { B } ) ph <-> ( A. x e. { A } ph /\ A. x e. { B } ph ) )
8 6 7 bitri 
 |-  ( A. x e. { A , B } ph <-> ( A. x e. { A } ph /\ A. x e. { B } ph ) )
9 1 3 ralsngf 
 |-  ( A e. V -> ( A. x e. { A } ph <-> ps ) )
10 2 4 ralsngf 
 |-  ( B e. W -> ( A. x e. { B } ph <-> ch ) )
11 9 10 bi2anan9 
 |-  ( ( A e. V /\ B e. W ) -> ( ( A. x e. { A } ph /\ A. x e. { B } ph ) <-> ( ps /\ ch ) ) )
12 8 11 bitrid 
 |-  ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } ph <-> ( ps /\ ch ) ) )