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Theorem mtp-xor 1536
 Description: Modus tollendo ponens (original exclusive-or version), aka disjunctive syllogism, one of the five "indemonstrables" in Stoic logic. The rule says, "if φ is not true, and either φ or ψ (exclusively) are true, then ψ must be true." Today the name "modus tollendo ponens" often refers to a variant, the inclusive-or version as defined in mtp-or 1538. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mpto2 1534, that is, it is exclusive-or df-xor 1305), rule 3 of [Sanford] p. 39 (where it is not as clearly stated which kind of "or" is used but it appears to be in the same sense as mpto2 1534), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 4-Jul-2016.) (Proof shortened by Wolf Lammen, 11-Nov-2017.)
Hypotheses
Ref Expression
mtp-xor.1 ¬ φ
mtp-xor.2 (φψ)
Assertion
Ref Expression
mtp-xor ψ

Proof of Theorem mtp-xor
StepHypRef Expression
1 mtp-xor.1 . . 3 ¬ φ
2 mtp-xor.2 . . . 4 (φψ)
3 xorneg 1313 . . . 4 ((¬ φ ⊻ ¬ ψ) ↔ (φψ))
42, 3mpbir 200 . . 3 φ ⊻ ¬ ψ)
51, 4mpto2 1534 . 2 ¬ ¬ ψ
65notnotri 106 1 ψ
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ⊻ wxo 1304 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-xor 1305 This theorem is referenced by: (None)
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