Propositional calculus - I can't get why the answer for this test question is what it is Consider the following premises.


If A = B then B = C.
B != C.
If C > D then D &lt; E.
F != G and A = B.
A = B or C > D.


Alternatives:

a) F != G

b) F != G and D &lt; E

c) A = B

d) B = C or D &lt; E

e) D &lt; E

The answer sheet says the correct alternative is B.

I am confused, because (1) and (2) seems to imply that A != B. In which case (4) would be contradictory. That's why I am wondering if the question itself is wrong.
I suspect that there is a typo on your source's part, namely being that (4) should be a disjunction. 

Making this correction, we have

(6) $A\neq B$. Take the contrapositive of (1) and applying modus ponens with (2).

(7) $C&gt;D$. Use disjunctive syllogism on (5) with (6).

(8) $D&lt;E$. Apply modus ponens on (3) with (7).

(9) $F\neq G$. Apply disjunctive syllogism on (modified 4) with (6).

(10) b) Conjunction between (8) and (9).

Thus b) would then indeed be a correct answer.
(1) and (2) do not imply that A != B. Indeed is an if, so that IF  A = B, then we have that A = B = C. However in 2, it is not stated that A=B, we only know that B != C,which implies that A != B.

To prove the equality we see that:
 from 4) F!=G and A=B, which implies 5) A=B or C > D.
Hence from 3) we get the answer b!

Notice that the passage from 5 to 3 works as 'or' works when one of the propositions or both are true, so we can take  C > D to be true.