Theorem mtord 784

Description: A modus tollens deduction involving disjunction. (Contributed by Jeff Hankins, 15-Jul-2009.) (Revised by Mario Carneiro, 31-Jan-2015.)

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.3	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃)))
2		mtord.1	. . . . 5 ⊢ (𝜑 → ¬ 𝜒)
3	2	pm2.21d 620	. . . 4 ⊢ (𝜑 → (𝜒 → ¬ 𝜓))
4		mtord.2	. . . . 5 ⊢ (𝜑 → ¬ 𝜃)
5	4	pm2.21d 620	. . . 4 ⊢ (𝜑 → (𝜃 → ¬ 𝜓))
6	3, 5	jaod 718	. . 3 ⊢ (𝜑 → ((𝜒 ∨ 𝜃) → ¬ 𝜓))
7	1, 6	syld 45	. 2 ⊢ (𝜑 → (𝜓 → ¬ 𝜓))
8	7	pm2.01d 619	1 ⊢ (𝜑 → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  swoer  6629