Consider the usual logical connectors $\wedge, \vee, \supset, \neg$ (i.e., "and", "or", material implication, negation) and the "stroke" $/$ defined as $p / q := (\neg p) \vee (\neg q)$.
In his book "Introduction to Logic: And to the Methodology of Deductive Sciences" (ISBN-10: 048628462X) the author A. Tarski presents on page 176 a section about "Closed systems of sentences" which runs as follows:
There exists a general logical law which permits us in some cases, when we have succeeded in proving several conditional propositions, to infer from the form of these propositions that the corresponding converse propositions may be also considered as proved.
Assume we are given a finite system of implications to which we will give the following schematic form $p \supset q, p' \supset q', p'' \supset q'', ...$
If the antecedents (premisses) exhaust all possible cases, that is, if it is true that $p \vee p' \vee p'' \vee ...$, and if simultaneously their consequents (conclusions) exclude one another (incompatibility) $q/q' \wedge q/q'' \wedge q'/q'' \wedge ...$, then the converse implications are entailed $q \supset p, q' \supset p', q'' \supset p'', ...$
Consider as example the following simple system of sentences: $((p \supset q) \wedge (p' \supset q') \wedge (p'' \supset q'') \wedge (p \vee p' \vee p'') \wedge (q/q') \wedge (q/q'') \wedge (q'/q'')) \supset ((q \supset p) \wedge (q' \supset p') \wedge (q'' \supset p''))$
My question is: How can the general claim in Tarski's book above be proved?