Theorem ex-natded5.8-2 30434

Description: A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 30433. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ex-natded5.8.1	⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃))
ex-natded5.8.2	⊢ (𝜑 → (𝜏 → 𝜃))
ex-natded5.8.3	⊢ (𝜑 → 𝜒)
ex-natded5.8.4	⊢ (𝜑 → 𝜏)

Assertion

Ref	Expression
ex-natded5.8-2	⊢ (𝜑 → ¬ 𝜓)

Proof of Theorem ex-natded5.8-2

Step	Hyp	Ref	Expression
1		ex-natded5.8.4	. . 3 ⊢ (𝜑 → 𝜏)
2		ex-natded5.8.2	. . 3 ⊢ (𝜑 → (𝜏 → 𝜃))
3	1, 2	mpd 15	. 2 ⊢ (𝜑 → 𝜃)
4		ex-natded5.8.3	. . 3 ⊢ (𝜑 → 𝜒)
5		ex-natded5.8.1	. . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃))
6	4, 5	mpan2d 694	. 2 ⊢ (𝜑 → (𝜓 → ¬ 𝜃))
7	3, 6	mt2d 136	1 ⊢ (𝜑 → ¬ 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395

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 207  df-an 396

This theorem is referenced by: (None)