It is easy to prove that an apple is an apple: The equation x=x is a tautology, for all x. So if x=apple, we can substitute it in the equation x=x and get apple=apple, so an apple is an apple.

But how can one prove (using standard deductive logic) that an apple is not an orange, since x!=y is not a tautology for all x,y?
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In mathematics, an equals sign connects subject and predicate.

The statement, apple=orange, is the same as the statement, an apple is an orange.

In order to determine an apple is an apple, you used the principle of identity. That which is the same is the same.

In order to determine an apple is not an orange, we use the principle of difference. That which is the same is not different, or two different things are not the same. 

General reasoning from sense infers that apples are not oranges. However, if we use the power of abstraction, we can take the concept of an apple, its "appleness", and a concept of an orange, its "orangeness". These conceptions are known though general reasoning from sense and are often informed from the rest of society, e.g. academia. 

An apple's "appleness" can only belong to apples. An orange's "orangeness" can only belong to oranges. "appleness" contains no "orangeness". Therefore, something containing "orangeness", namely an orange, is never something containing "appleness", namely an apple, i.e. apples are never oranges.
the logic used here is mathematical logic . but a mathematical logic is only accepted if there is no way to disprove the given logic i.e to say 2+3 =5 and 3+2 also=5 assures the logic that counting 5 pieces from left or right will always remain same because we cant find any two no's which dont obey this logic .
so if we assume apple as x and orange as y such that we know always x=x and y=y is true(because any experiment cant disprove this .an apple remains an apple forever)
now if somehow apple could change into orange after a few years then x=x would have been wrong and then x=y would hold good
also falsness of only x=x directly assures the validity x=y.

but validity of x=x doesnt directly assures the falsness of x=y
note:apple=apple means apple will never change to anything else (including orange)
and orange=orange means orange will never change to anything else (including apple)
so both the chances are finished and there is no other way by which we can say that apple = orange.if one of the two statements x=x and y=y becomes false then x=y can be true means both x=x and y=y should be true