Theorem mtp-xor 1313

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 1314. See rule 3 on [Lopez-Astorga] p. 12 (note that the "or" is the same as mpto2 1312, that is, it is exclusive-or df-xor 1266), 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 1312), 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	⊢ (φ ⊻ ψ)

Assertion

Ref	Expression
mtp-xor	⊢ ψ

Proof of Theorem mtp-xor

Step	Hyp	Ref	Expression
1		mtp-xor.1	. 2 ⊢ ¬ φ
2		mtp-xor.2	. . . . 5 ⊢ (φ ⊻ ψ)
3		df-xor 1266	. . . . 5 ⊢ ((φ ⊻ ψ) ↔ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ)))
4	2, 3	mpbi 133	. . . 4 ⊢ ((φ ∨ ψ) ∧ ¬ (φ ∧ ψ))
5	4	simpli 104	. . 3 ⊢ (φ ∨ ψ)
6	5	ori 641	. 2 ⊢ (¬ φ → ψ)
7	1, 6	ax-mp 7	1 ⊢ ψ

Syntax hints:  ¬ wn 3   ∧ wa 97   ∨ wo 628   ⊻ wxo 1265

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 545  ax-io 629

This theorem depends on definitions:  df-bi 110  df-xor 1266

This theorem is referenced by: (None)