Metamath Proof Explorer
Description: Deduction for inequality. (Contributed by NM, 25Oct1999) (Proof
shortened by Wolf Lammen, 19Nov2019)


Ref 
Expression 

Hypothesis 
neeq1d.1 
$${\u22a2}{\phi}\to {A}={B}$$ 

Assertion 
neeq1d 
$${\u22a2}{\phi}\to \left({A}\ne {C}\leftrightarrow {B}\ne {C}\right)$$ 
Proof
Step 
Hyp 
Ref 
Expression 
1 

neeq1d.1 
$${\u22a2}{\phi}\to {A}={B}$$ 
2 
1

eqeq1d 
$${\u22a2}{\phi}\to \left({A}={C}\leftrightarrow {B}={C}\right)$$ 
3 
2

necon3bid 
$${\u22a2}{\phi}\to \left({A}\ne {C}\leftrightarrow {B}\ne {C}\right)$$ 