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Metamath Proof Explorer

Theorem rextpg

Description: Convert a restricted existential quantification over a triple to a disjunction. (Contributed 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 rextpg 
|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( E. 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 rexprg 
 |-  ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )
5 4 orbi1d 
 |-  ( ( A e. V /\ B e. W ) -> ( ( E. x e. { A , B } ph \/ E. x e. { C } ph ) <-> ( ( ps \/ ch ) \/ E. x e. { C } ph ) ) )
6 3 rexsng 
 |-  ( C e. X -> ( E. x e. { C } ph <-> th ) )
7 6 orbi2d 
 |-  ( C e. X -> ( ( ( ps \/ ch ) \/ E. x e. { C } ph ) <-> ( ( ps \/ ch ) \/ th ) ) )
8 5 7 sylan9bb 
 |-  ( ( ( A e. V /\ B e. W ) /\ C e. X ) -> ( ( E. x e. { A , B } ph \/ E. x e. { C } ph ) <-> ( ( ps \/ ch ) \/ th ) ) )
9 8 3impa 
 |-  ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( E. x e. { A , B } ph \/ E. x e. { C } ph ) <-> ( ( ps \/ ch ) \/ th ) ) )
10 df-tp 
 |-  { A , B , C } = ( { A , B } u. { C } )
11 10 rexeqi 
 |-  ( E. x e. { A , B , C } ph <-> E. x e. ( { A , B } u. { C } ) ph )
12 rexun 
 |-  ( E. x e. ( { A , B } u. { C } ) ph <-> ( E. x e. { A , B } ph \/ E. x e. { C } ph ) )
13 11 12 bitri 
 |-  ( E. x e. { A , B , C } ph <-> ( E. x e. { A , B } ph \/ E. x e. { C } ph ) )
14 df-3or 
 |-  ( ( ps \/ ch \/ th ) <-> ( ( ps \/ ch ) \/ th ) )
15 9 13 14 3bitr4g 
 |-  ( ( A e. V /\ B e. W /\ C e. X ) -> ( E. x e. { A , B , C } ph <-> ( ps \/ ch \/ th ) ) )