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Theorem ex-natded5.13-2 27584
 Description: A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 27583. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ex-natded5.13.1 (𝜑 → (𝜓𝜒))
ex-natded5.13.2 (𝜑 → (𝜓𝜃))
ex-natded5.13.3 (𝜑 → (¬ 𝜏 → ¬ 𝜒))
Assertion
Ref Expression
ex-natded5.13-2 (𝜑 → (𝜃𝜏))

Proof of Theorem ex-natded5.13-2
StepHypRef Expression
1 ex-natded5.13.1 . 2 (𝜑 → (𝜓𝜒))
2 ex-natded5.13.2 . . 3 (𝜑 → (𝜓𝜃))
3 ex-natded5.13.3 . . . 4 (𝜑 → (¬ 𝜏 → ¬ 𝜒))
43con4d 114 . . 3 (𝜑 → (𝜒𝜏))
52, 4orim12d 919 . 2 (𝜑 → ((𝜓𝜒) → (𝜃𝜏)))
61, 5mpd 15 1 (𝜑 → (𝜃𝜏))
 Colors of variables: wff setvar class 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 197  df-or 384  df-an 385 This theorem is referenced by: (None)
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