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 ( 𝜑𝜓 )
Assertion mptxor ¬ 𝜓

Proof

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