A metatheorem in mathematical logic also known under the name "conditional proof." It states that if the sentential formula can be derived from the set of sentential formulas , then the sentential formula can be derived from .

In a less formal setting, this means that if a thesis can be proven under the hypotheses , then one can prove that implies under hypothesis .

This entry contributed by Margherita Barile

Kleene, S. C. "The Deduction Theorem." §21 in Introduction to Metamathematics. Princeton, NJ: Van Nostrand, pp. 90-94, 1964.

Monk, D. J. Mathematical Logic. New York: Springer-Verlag, p. 118, 1976.

Robbin, J. W. "The Deduction Theorem." §21 in Mathematical Logic. New York: W. A. Benjamin, pp. 16-20, 1969.

Shoenfield, J. R. "The Deduction Theorem." §3.3 in Mathematical Logic. Reading, MA: Addison-Wesley, pp. 33-34, 1967.

CITE THIS AS:

Barile, Margherita. "Deduction Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/DeductionTheorem.html