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Metamath Proof Explorer

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 )