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Suppose that we define:

"The president of USA" := John

Is it correct to say that when we have a statement like:

"The president of USA plays football".

we can expand it like:

"John plays football".

?

In general, are we allowed to make such definitions where on the left hand side we have a sequence of words?

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    Not necessarily so in your described case as such substitution rule of free variables once you have equality relation defined, aka, principle of indiscernibility of identicals, depends on context. If your context has referential opacity then your substitution result may not reflect your intention. For example, the true statement of "John" has four letters is false under your LHS substitution. Another example is 8 is necessarily greater than 5 is false under substitution of 8 with "number of planets" with the usual de dicto reading...
    – Double Knot
    Jul 23, 2022 at 18:58

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In general, are we allowed to make such definitions where on the left hand side we have a sequence of words?

Yes:

  • Names
  • Synonyms

"The president of USA plays football".

we can expand it like:

"John plays football".

This is not a good example. The President of the USA is a societal role – a title, if you will. That phrase refers to the current president; in this sentence, the phrase means John, but in a sentence 100 years later, it'll mean somebody else.

You are, of course, allowed to define any word (or sequence of words) to mean anything you like, but defining words differently to how other people define them does not aid in communication.

Humpty Dumpty: When I use a word, it means just what I choose it to mean. Neither more nor less.
Alice: I wonder what all those words you just said meant. Maybe you're telling me I can have all your stuff!
Humpty Dumpty: What!? No!
Alice: Your car, too? Gosh, thanks!

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I presume that the question is related to the theory of quotation and shall set down several remarks that might be of help approaching the issue.

For technical matters involving formal languages, usually single quotation marks are used, and double quotation marks are mostly reserved to the verbal quotations, as for example,

  • Marx says "Alles gesellschaftliche Leben ist wesentlich praktisch."

So, I'll stick to single quotation marks. As for the cases concerning the punctuation of a natural language, no doubt, the conventions of that language should be employed.

As a general principle, when an expression is mentioned as opposed to used, it is enclosed between quotation marks. Hence, as a philosophical question, quotation is most related to use-mention distinction. However, discrimination of use and mention cases is a matter of not absolute, but a relative matter. To put crudely, within a particular discourse, we identify an object level and a meta-level, then decide on when an expression is used (at object level) or mentioned (at meta-level).

But, on the one hand, the cases in actual practice of language are not so clear-cut, using and mentioning are complex linguistic devices, furthermore, quotation and mention are not equivalent (a sentence may well be mentioned without quoted, etc.). On the other hand, a hierarchy of use-mention instances can readily occur and get tangled:

In sentence 1, the expression E is used;

Sentence 2 mentions the expression E used in sentence 1;

Sentence 3 uses the mention in sentence 2 of the expression E;

. . .

Last but not least, an expression can be mentioned without quoted (viz. enclosed between quotation marks), hence, there is not a one-to-one between mention and quotation.

For now, I suppose, it is sufficient to be aware of such problematic facets of quotation and leave its ramifications to separate questions. For the present question, let us go over some examples and draw distinctions from them (the statements succeeding the examples are those that can be said on the basis of the example):

  • 'The president of USA' := John

John is a sentence with 17 letters and 3 spaces.

  • 'The president of USA' := 'John'

In each case the string of symbols 'The president of USA' occurs, it is replaceable with the string of symbols 'John'.

  • The president of USA := John

The sentences 'The president of USA plays football' and 'John plays football' are referentially equivalent.

  • (The name) 'John' designates the president of US.

Neither the person, nor the string of symbols, but the name 'John' (a noun of the language) refers to the president of USA (notice the conflicting situation: why not the string of symbols, but the name? That is why sometimes the phrase 'the name' is added to clarify the intended meaning).

  • ⌜Prn is the president of USA⌝ is a sentence with a pronoun in the subject position.

Assuming Prn is a variable, other pronouns can legitimately replace it. The sentences that can immediately follow from the given one:

  1. 'He is the president of USA' is a sentence with a pronoun in the subject position.
  2. 'Someone is the president of USA' is a sentence with a pronoun in the subject position.
  3. 'She is the president of USA' is a sentence with a pronoun in the subject position.
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