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Theorem ex-natded5.8-2 28202
 Description: A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 28201. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ex-natded5.8.1 (𝜑 → ((𝜓𝜒) → ¬ 𝜃))
ex-natded5.8.2 (𝜑 → (𝜏𝜃))
ex-natded5.8.3 (𝜑𝜒)
ex-natded5.8.4 (𝜑𝜏)
Assertion
Ref Expression
ex-natded5.8-2 (𝜑 → ¬ 𝜓)

Proof of Theorem ex-natded5.8-2
StepHypRef Expression
1 ex-natded5.8.4 . . 3 (𝜑𝜏)
2 ex-natded5.8.2 . . 3 (𝜑 → (𝜏𝜃))
31, 2mpd 15 . 2 (𝜑𝜃)
4 ex-natded5.8.3 . . 3 (𝜑𝜒)
5 ex-natded5.8.1 . . 3 (𝜑 → ((𝜓𝜒) → ¬ 𝜃))
64, 5mpan2d 693 . 2 (𝜑 → (𝜓 → ¬ 𝜃))
73, 6mt2d 138 1 (𝜑 → ¬ 𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399 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 210  df-an 400 This theorem is referenced by: (None)
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