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	⊢ (φ ⊻ ψ)

Assertion

Ref	Expression
mtp-xor	⊢ ψ

Proof of Theorem mtp-xor

Step	Hyp	Ref	Expression
1		mtp-xor.1	. . 3 ⊢ ¬ φ
2		mtp-xor.2	. . . 4 ⊢ (φ ⊻ ψ)
3		xorneg 1313	. . . 4 ⊢ ((¬ φ ⊻ ¬ ψ) ↔ (φ ⊻ ψ))
4	2, 3	mpbir 200	. . 3 ⊢ (¬ φ ⊻ ¬ ψ)
5	1, 4	mpto2 1534	. 2 ⊢ ¬ ¬ ψ
6	5	notnotri 106	1 ⊢ ψ

Syntax hints:  ¬ wn 3   ⊻ wxo 1304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-xor 1305

This theorem is referenced by: (None)