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Metamath Proof Explorer

Theorem rexprg

Description: Convert a restricted existential quantification over a pair to a disjunction. (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 rexprg 
|- ( ( A e. V /\ B e. W ) -> ( E. 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 1 notbid 
 |-  ( x = A -> ( -. ph <-> -. ps ) )
4 2 notbid 
 |-  ( x = B -> ( -. ph <-> -. ch ) )
5 3 4 ralprg 
 |-  ( ( A e. V /\ B e. W ) -> ( A. x e. { A , B } -. ph <-> ( -. ps /\ -. ch ) ) )
6 ralnex 
 |-  ( A. x e. { A , B } -. ph <-> -. E. x e. { A , B } ph )
7 pm4.56 
 |-  ( ( -. ps /\ -. ch ) <-> -. ( ps \/ ch ) )
8 6 7 bibi12i 
 |-  ( ( A. x e. { A , B } -. ph <-> ( -. ps /\ -. ch ) ) <-> ( -. E. x e. { A , B } ph <-> -. ( ps \/ ch ) ) )
9 notbi 
 |-  ( ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) <-> ( -. E. x e. { A , B } ph <-> -. ( ps \/ ch ) ) )
10 8 9 sylbb2 
 |-  ( ( A. x e. { A , B } -. ph <-> ( -. ps /\ -. ch ) ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )
11 5 10 syl 
 |-  ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } ph <-> ( ps \/ ch ) ) )