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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 .
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.
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