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Theorem mtp-xor 1299
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 1300. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mpto2 1298, that is, it is exclusive-or df-xor 1252), 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 1298), and rule A5 in [Hitchcock] p. 5 (exclusive-or is expressly used). (Contributed by David A. Wheeler, 2-Mar-2018.)
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 . 2 ¬ φ
2 mtp-xor.2 . . . . 5 (φψ)
3 df-xor 1252 . . . . 5 ((φψ) ↔ ((φ ψ) ¬ (φ ψ)))
42, 3mpbi 133 . . . 4 ((φ ψ) ¬ (φ ψ))
54simpli 104 . . 3 (φ ψ)
65ori 629 . 2 φψ)
71, 6ax-mp 7 1 ψ
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   wo 616  wxo 1251
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 533  ax-io 617
This theorem depends on definitions:  df-bi 110  df-xor 1252
This theorem is referenced by: (None)
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