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sometimes one sees/reads assertions such as "[the bounded inverse theorem] is equivalent to both the open mapping theorem and the closed graph theorem", but taken formally and literally this would amount to the trivial observation that (φ∧ψ) → (φ↔ψ) [ which holds in logics weaker than classical logic ], and, in particular, that all theorems of a given theory are equivalent, which does not seem to be the intended meaning

the historical examples of euclidean geometry and set theory suggest that, from a slightly technical viewpoint, it would be that "theorems φ and ψ of a given theory Γ are 'equivalent' just in case there is a subtheory Γ' of Γ such that Γ'⊬φ, and Γ'⊬ψ, but Γ'⊢(φ↔ψ)" [think of euclid and playfair's axioms, for the first, and choice, well-ordering and zorn, for the latter], and this also seems to be the approach of reverse mathematics, but it may not necessarily be the naïve use/viewpoint of mathematicians not versed in matters of logic, as it seems to hint more at some intuitive grasp of the 'content' of the theorems/statements than to provability in such more strict terms

is this use/notion of 'equivalence' discussed/examined somewhere in the philosophy of mathematics literature (perhaps in the 'philosophy of mathematical practice' literature), or perhaps even in the usual mathematics literature?

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    I'd say it means φ is "easily derivable" from ψ and vice versa. This could be interpreted as saying: starting from φ and a set of "common" theorems, it is possible to prove ψ in a small number of deduction steps, and it is not possible to prove ψ in a small number of deduction steps if we omit φ from the premises, and the reverse direction also holds. This is an inherently vague concept (how many steps? which are the theorems taken as common?).
    – causative ♦
    Commented May 5 at 23:57
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    @ac15 Working over a weak base theory (a la reverse mathematics) does improve things to a certain extent; however, we're still left with examples where the intuitive sense of "sameness" doesn't match the reverse-mathematical one (see e.g. Timothy Chow's answer here; more generally, that entire thread is relevant). I think there are ultimately several notions of equivalence at play, some of which are well-understood enough to be formalized and others of which definitely aren't (yet!).
    – Noah Schweber
    Commented May 6 at 0:11
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    @ac15 I think Timothy's example is really serious FWIW. Of course mileage will vary on this point, but I do believe Sperner/Brouwer is a great example of how the revmath picture doesn't always capture what many/most mathematicians see as the "right" dividing lines/equivalences. (To be honest, I'm a bit ashamed that my own answer there got accepted; it toes the revmath party line a bit too neatly in retrospect.)
    – Noah Schweber
    Commented May 6 at 0:20
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    @Conifold, you wrote: "The Γ' approach works well for axioms rather than theorems." However, from the viewpoint of reverse mathematics, the distinction often turns out to be an illusory one. This is somehow the main thrust of reverse mathematics a la Simpson and others.
    – Mikhail Katz
    Commented May 6 at 8:23
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    @MauroALLEGRANZA, Playfair's axiom happens to be an example that was mentioned in the body of the question, so it is not clear what you propose to add here.
    – Mikhail Katz
    Commented May 6 at 9:10

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There is an active and (somewhat) fashionable field called Reverse Mathematics that gives a precise meaning to such equivalence. A few decades of progress have revealed that most theorems of, for example, classical analysis fall into five groups of increasing strength, which we will label A,B,C,D,E to avoid notation that's too technical. Here A is a fairly finitistic theory which is somehow "the most basic" one, B is stronger than A, etc. Already at the level of the theory B, there are several impressive classical results that are equivalent among themselves over the base theory A. Here is a relevant quote from Simpson. Simpson writes on page 378 of his Subsystems of Second Order Arithmetic (here B=WKL_0):

"For example, all of the following key theorems of infinitistic mathematics are provable in WKL_0 and therefore, by theorem IX.3.16, reducible to finitism. (1) The Heine/Borel covering theorem for closed bounded subsets of R^n or for closed subsets of any compact metric space. (2) Basic properties of continuous real-valued functions of several real variables. (3) The local existence theorem for solutions of ordinary differential equations. (4) The Hahn/Banach theorem in separable Banach spaces. (5) The existence theorem for prime ideals in countable commutative rings. (6) Existence and uniqueness of the algebraic closure of a countable field. (7) Orderability and existence of the real closure of a countable formally real field."

When one passes to the stronger theories C, D, and E, one gets even more impressive results (and equivalences among them in the precise sense explained above).

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