Theorem mtord

Description: A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009)

	Ref	Expression
Hypotheses	mtord.1	⊢ ( 𝜑 → ¬ 𝜒 )
	mtord.2	⊢ ( 𝜑 → ¬ 𝜃 )
	mtord.3	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ∨ 𝜃 ) ) )
Assertion	mtord	⊢ ( 𝜑 → ¬ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		mtord.1	⊢ ( 𝜑 → ¬ 𝜒 )
2		mtord.2	⊢ ( 𝜑 → ¬ 𝜃 )
3		mtord.3	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 ∨ 𝜃 ) ) )
4		pm2.53	⊢ ( ( 𝜒 ∨ 𝜃 ) → ( ¬ 𝜒 → 𝜃 ) )
5	3 1 4	syl6ci	⊢ ( 𝜑 → ( 𝜓 → 𝜃 ) )
6	2 5	mtod	⊢ ( 𝜑 → ¬ 𝜓 )