Conditional Proof Walkthrough
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Conditional proof assumes the antecedent of the conditional it wants, derives the consequent under that assumption, then discharges the assumption to produce the conditional as a line that depends on the premises alone.
Premises: A horseshoe B, and B horseshoe C. Prove A horseshoe C, without using hypothetical syllogism. Line three: assume A, and indent from here. The indentation is a fence, and everything inside it holds only under the assumption. Line four: B, from lines one and three by modus ponens. Line five: C, from lines two and four by modus ponens. The consequent has been reached, so the assumption has done its work. Discharge it. Line six: A horseshoe C, citing lines three through five as a conditional proof, and the indentation closes. Line six sits outside the fence, which is what makes it usable — it rests on the two premises and not on the assumption, because the assumption is now recorded inside the conditional rather than standing behind it.
Lines inside the fence cannot be cited after it closes. They were derived under an assumption that is no longer in force, and reaching back into a discharged scope produces a line resting on something the proof has withdrawn — which is the most common way a conditional proof goes wrong while still looking orderly.