comma or semicolon before "that is" and repeating 'that'

In the following two sentences

1. We need to show that Phrase; that is, that Phrase A'.
2. We need to show that Phrase A, that is, Phrase A'.

Which one is correct or more natural?

Phrase A and Phrase A' are two sentences that express the same thing. That they are the same are obvious to see and is not our concern. Add an example for sentence 1. (We are not going to show that "a+b=c is the same as a=c-b", which is trivial.)

We need to show that a+b=c; that is, that a=c-b.

Thanks a lot.

Solution 1:

Your new example now contains a main verb (=, that is, equals), so what you have is correct.

But you can use either a semicolon or commas.

To use a "simpler" sentence:

1 We need to show that 2 plus 2 equals 4; that is, that 4 is the sum of 2 and 2.

or

2 We need to show that 2 plus 2, that is, the sum of 2 and 2, is [equals] 4.

Note the placement of the punctuation and of "that is".

In 1, we have two separate clauses, which we can join with a semicolon; the latter clause is clarifying, and can be removed without making the sentence ungrammatical.

In 2, we have a parenthetical statement. This is also clarifying and can also be removed without affecting the grammaticality or meaning.

A parenthetical statement can also use dashes or parentheses:

2a We need to show that 2 plus 2 – that is, the sum of 2 and 2 – is 4.

or

2b We need to show that 2 plus 2 (that is, the sum of 2 and 2) is 4.

Note that I have used en dashes in 2a; unspaced em dashes are generally preferred in American English.