Theorem mtord 690

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.1	. . 3 ⊢ (𝜑 → ¬ 𝜒)
3		mtord.3	. . . 4 ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))
4		df-or 384	. . . 4 ⊢ ((𝜒 ∨ 𝜃) ↔ (¬ 𝜒 → 𝜃))
5	3, 4	syl6ib 240	. . 3 ⊢ (𝜑 → (𝜓 → (¬ 𝜒 → 𝜃)))
6	2, 5	mpid 43	. 2 ⊢ (𝜑 → (𝜓 → 𝜃))
7	1, 6	mtod 188	1 ⊢ (𝜑 → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 196  df-or 384

This theorem is referenced by:  swoer  7659  inar1  9476