Let us suppose that the following identity statement is true

(1) Byron = Arnold

Suppose further

(2) Byron is the originator of Frisbianity

It follows that

(3) Arnold is the originator of Frisbianity

Likewise from

Byron studied in New Haven

it follows that

Arnold studied in New Haven

It also follows that

If Byron enjoys a good painting, then Arnold enjoys a good painting

and

If Byron’s German needs work, then Arnold’s German needs work

And so on. Whatever is truthfully predicated of Byron is also true of Arnold.

Of course the converse is also true. Whatever is truthfully predicated of Arnold is also true of Byron. Again, assuming the truth of (1).

Arnold was a member of Diatheke Church

and

Arnold taught at NCHS

imply respectively,

Byron was a member of Diatheke Church

and

Byron taught at NCHS

Suppose

(4) Arnold ran off with his pastor’s wife

and

(5) Arnold was excommunicated from the OPC

We may conclude, assuming (1),

(6) Byron ran off with his pastor’s wife

(7) Byron was excommunicated from the OPC

There are problems with applying Leibniz’s law in certain contexts. Suppose the following is true.

Sean believes (2)

We may not conclude from this:

Sean believes (3)

For though we have stipulated the truth of (1), it may not be the case that

(8) Sean believes (1)

The lesson from this is that while Leibniz’s law applies in the predicate calculus, it does not always apply in contexts where propositions are embedded within beliefs.

Likewise from

Sean believes (4)

we cannot infer

Sean believes (6)

unless (8) is true.

Of course if (8) and

Sean believes (5)

were true, so would

Sean believes (7)

For further discussion about propositional attitudes see here.