Theorem mptxor 1773

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.) (Proof shortened by BJ, 19-Apr-2019.)

Hypotheses

Ref	Expression
mptxor.min	⊢ 𝜑
mptxor.maj	⊢ (𝜑 ⊻ 𝜓)

Assertion

Ref	Expression
mptxor	⊢ ¬ 𝜓

Proof of Theorem mptxor

Step	Hyp	Ref	Expression
1		mptxor.min	. 2 ⊢ 𝜑
2		mptxor.maj	. . 3 ⊢ (𝜑 ⊻ 𝜓)
3		xornan 1512	. . 3 ⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ∧ 𝜓))
4	2, 3	ax-mp 5	. 2 ⊢ ¬ (𝜑 ∧ 𝜓)
5	1, 4	mptnan 1772	1 ⊢ ¬ 𝜓

Syntax hints:  ¬ wn 3   ∧ wa 395   ⊻ wxo 1503

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 206  df-an 396  df-or 844  df-xor 1504

This theorem is referenced by: (None)