Theorem mpto2 1534

Description: Modus ponendo tollens 2, one of the "indemonstrables" in Stoic logic. Note that this uses exclusive-or ⊻. See rule 2 on [Lopez-Astorga] p. 12 , rule 4 on [Sanford] p. 39 and rule A4 in [Hitchcock] p. 5 . (Contributed by David A. Wheeler, 3-Jul-2016.) (Proof shortened by Wolf Lammen, 12-Nov-2017.)

Hypotheses

Ref	Expression
mpto2.1	⊢ φ
mpto2.2	⊢ (φ ⊻ ψ)

Assertion

Ref	Expression
mpto2	⊢ ¬ ψ

Proof of Theorem mpto2

Step	Hyp	Ref	Expression
1		mpto2.1	. 2 ⊢ φ
2		mpto2.2	. . . 4 ⊢ (φ ⊻ ψ)
3		df-xor 1305	. . . 4 ⊢ ((φ ⊻ ψ) ↔ ¬ (φ ↔ ψ))
4	2, 3	mpbi 199	. . 3 ⊢ ¬ (φ ↔ ψ)
5		xor3 346	. . 3 ⊢ (¬ (φ ↔ ψ) ↔ (φ ↔ ¬ ψ))
6	4, 5	mpbi 199	. 2 ⊢ (φ ↔ ¬ ψ)
7	1, 6	mpbi 199	1 ⊢ ¬ ψ

Syntax hints:  ¬ wn 3   ↔ wb 176   ⊻ 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:  mtp-xor  1536