Question 626372
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Assume for contradiction that B&#8800;C 

Then either B&#8840;C or C&#8840;B 

We only need to disprove one of these since after we have 
disproved one of them, we can disprove the other just by 
swapping the roles of B and C.

We will assume B&#8840;C

Then &#8707;x such that x&#8712;B and x&#8713;C

The either x&#8712;A or x&#8713;A

Case 1: x&#8712;A.  Then since x&#8712;B, x&#8712;A&#8898;B. But
since x&#8713;C, x&#8713;A&#8898;C. Therefore A&#8898;B&#8800;A&#8898;C, a
contadiction since A&#8898;B=A&#8898;C is given. So case 1 is disproved.

Case 2: x&#8713;A. Then since x&#8712;B, x&#8712;A&#8899;B. But
since x&#8713;C, x&#8713;A&#8899;C. Therefore A&#8899;B&#8800;A&#8899;C, a
contadiction, since A&#8899;B=A&#8899;C is given. So case 2 is disproved.
   
Therefore B&#8840;C is false and B&#8838;C is true.

By swapping the roles of B and C in the above, C&#8840;B is false 
and C&#8838;B is true.

Therefore B=C.

Edwin</pre>