Theorem mtord 892

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

Hypotheses

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

Assertion

Ref	Expression
mtord	⊢ (𝜑 → ¬ 𝜓)

Proof of Theorem mtord

Step	Hyp	Ref	Expression
1		mtord.2	. 2 ⊢ (𝜑 → ¬ 𝜃)
2		mtord.3	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))
3		mtord.1	. . 3 ⊢ (𝜑 → ¬ 𝜒)
4		pm2.53 864	. . 3 ⊢ ((𝜒 ∨ 𝜃) → (¬ 𝜒 → 𝜃))
5	2, 3, 4	syl6ci 72	. 2 ⊢ (𝜑 → (𝜓 → 𝜃))
6	1, 5	mtod 201	1 ⊢ (𝜑 → ¬ 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 860

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-or 861

This theorem is used by:  ecase2d  1047  swoer  8722  inar1  10764  rtprmirr  26934