Theorem mpto1 1533

Description: Modus ponendo tollens 1, one of the "indemonstrables" in Stoic logic. See rule 1 on [Lopez-Astorga] p. 12 , rule 1 on [Sanford] p. 40, and rule A3 in [Hitchcock] p. 5. Sanford describes this rule second (after mpto2 1534) as a "safer, and these days much more common" version of modus ponendo tollens because it avoids confusion between inclusive-or and exclusive-or. (Contributed by David A. Wheeler, 3-Jul-2016.)

Hypotheses

Ref	Expression
mpto1.1	⊢ φ
mpto1.2	⊢ ¬ (φ ∧ ψ)

Assertion

Ref	Expression
mpto1	⊢ ¬ ψ

Proof of Theorem mpto1

Step	Hyp	Ref	Expression
1		mpto1.1	. 2 ⊢ φ
2		mpto1.2	. . 3 ⊢ ¬ (φ ∧ ψ)
3	2	imnani 412	. 2 ⊢ (φ → ¬ ψ)
4	1, 3	ax-mp 5	1 ⊢ ¬ ψ

Syntax hints:  ¬ wn 3   ∧ wa 358

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-an 360

This theorem is referenced by: (None)