Some Restricted Lindenbaum Theorems Equivalent to the Axiom of Choice

Dzik (1981) gives a direct proof of the axiom of choice from the generalized Lindenbaum extension theorem LET. The converse is part of every decent logical education. Inspection of Dzik’s proof shows that its premise let attributes a very special version of the Lindenbaum extension property to a very special class of deductive systems. The problem therefore arises of giving a direct proof, not using the axiom of choice, of the conditional let ⇒ LET. A partial solution is provided.