Theorem mptnan 1765

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 mptxor 1766) 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
mptnan.min	⊢ 𝜑
mptnan.maj	⊢ ¬ (𝜑 ∧ 𝜓)

Assertion

Ref	Expression
mptnan	⊢ ¬ 𝜓

Proof of Theorem mptnan

Step	Hyp	Ref	Expression
1		mptnan.min	. 2 ⊢ 𝜑
2		mptnan.maj	. . 3 ⊢ ¬ (𝜑 ∧ 𝜓)
3	2	imnani 403	. 2 ⊢ (𝜑 → ¬ 𝜓)
4	1, 3	ax-mp 5	1 ⊢ ¬ 𝜓

Syntax hints:  ¬ wn 3   ∧ wa 398

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 209  df-an 399

This theorem is referenced by:  mptxor  1766  alephsucpw2  9531  aleph1re  15592