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Metamath Proof Explorer

Theorem mptxor

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)

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

Proof

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