Description: The ratio of nonzero numbers is nonzero. (Contributed by NM, 2-Aug-2004) (Proof shortened by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
---|---|---|---|
Assertion | divne0b | |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A =/= 0 <-> ( A / B ) =/= 0 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | diveq0 | |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( ( A / B ) = 0 <-> A = 0 ) ) |
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2 | 1 | bicomd | |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A = 0 <-> ( A / B ) = 0 ) ) |
3 | 2 | necon3bid | |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A =/= 0 <-> ( A / B ) =/= 0 ) ) |