Metamath Proof Explorer

Theorem annotanannot

Description: A conjunction with a negated conjunction. (Contributed by AV, 8-Mar-2022) (Proof shortened by Wolf Lammen, 1-Apr-2022)

Ref Expression
Assertion annotanannot 
|- ( ( ph /\ -. ( ph /\ ps ) ) <-> ( ph /\ -. ps ) )

Proof

Step Hyp Ref Expression
1 ibar 
 |-  ( ph -> ( ps <-> ( ph /\ ps ) ) )
2 1 bicomd 
 |-  ( ph -> ( ( ph /\ ps ) <-> ps ) )
3 2 notbid 
 |-  ( ph -> ( -. ( ph /\ ps ) <-> -. ps ) )
4 3 pm5.32i 
 |-  ( ( ph /\ -. ( ph /\ ps ) ) <-> ( ph /\ -. ps ) )