References for "equational" calculi for specific propositional logics.

I am interested in references for equational proof calculi (in the sense described below) for specific propositional logics. I am especially interested in references that deal with mundane or practical aspects of using these proof calculi such as proof length, or their suitability for by-hand proofs or computer-generated or computer-verified proofs.

My main motivation for wanting something like this is to get a family of proof calculi that is as flexible as the family of tableau calculi while being easier to type on a computer and taking up less space overall.

Two ideas that I heard about on this site has stuck with me over the years:

1. The system of logic based on Copi's rules as used in this question, for example with 19 rules, some of which are traditional inference rules and others are rules of replacement with a more equational / algebraic flavor.
2. Blok and Pigozzi's short book Algebraizable Logics.

Copi's rules are interesting because that system is not actually a proof calculus for classical propositional logic, but rather for something like Bochvar's logic because it doesn't have any theorems. You can fix this by allowing substitution instances of $a \to a$ as axioms, or by allowing use of the deduction theorem and its converse.

Algebraizable Logics is very abstract and is mostly interested meta-level theorems about classes of consequence relations, although they do examine particular consequence relations (i.e. particular logics) here and there.

It's not too much of a leap to wonder about proof calculi entirely based on rules of replacement / equations for specific propositional calculi.

Here is an ad hoc equational proof calculus for classical propositional logic. I have made no attempt to be minimal. Some of these identities can be derived from the others.

1. $ a \to a \approx x \to x $
2. $ a \to b \to c \approx b \to a \to c $
3. $ a \to c \approx a \to a \to c $
4. $ c \approx (a \to a) \to c $
5. $ a \to a \approx b \to (a \to a)$
6. $ \lnot\lnot a \approx a $
7. $ (a \to b) \to (\lnot b \to \lnot a)$
8. $ a \to (a \to b) \to c \approx a \to b \to c$

As proof of soundness, I note that all of these identities hold in the two element boolean algebra with $\to$ interpreted as classical implication and $\approx$ interpreted as the classical biconditional.

As proof of completeness, I prove the Łukasiewicz axioms for classical propositional calculus and the inference rule modus ponens.

0. Modus Ponens

I insist that a premise $\varphi$ is true by using $\varphi \approx \varphi \to \varphi$ (or $\varphi \approx \psi \to \psi$, which is equivalent by equation 1).

$$ \text{H1:}\;\; a \to b \approx (a \to b) \to (a \to b) \\ \text{H2:}\;\; a \approx (a \to a) \\ (a \to a) \to b \approx ((a \to a) \to b) \to ((a \to a) \to b) \\ b \approx b \to b$$

1. Axiom 1: $a \to b \to a$

$$ a \to a \approx (x \to x) \\ b \to (a \to a) \approx (x \to x) \\ a \to b \to a \approx (x \to x) $$

2. Axiom 2: $(a \to b \to c) \to (a \to b) \to a \to c$

$$ (a \to b \to c) \to (a \to b) \to a \to c \approx \\ (b \to a \to c) \to (a \to b) \to a \to c \approx \\ (b \to a \to c) \to b \to a \to c \approx \\ (a \to c) \to b \to a \to c \approx \\ c \to b \to a \to c \approx \\ b \to a \to c \to c \approx \\ c \to c$$

3. Axiom 3: $(\lnot a \to \lnot b) \to b \to a$

$$ (\lnot a \to \lnot b) \to b \to a \approx \\ (b \to a) \to b \to a $$