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Theorem ex-natded5.2-2 26448
Description: A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 26447 and ex-natded5.2i 26449. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ex-natded5.2.1 (𝜑 → ((𝜓𝜒) → 𝜃))
ex-natded5.2.2 (𝜑 → (𝜒𝜓))
ex-natded5.2.3 (𝜑𝜒)
Assertion
Ref Expression
ex-natded5.2-2 (𝜑𝜃)

Proof of Theorem ex-natded5.2-2
StepHypRef Expression
1 ex-natded5.2.3 . . 3 (𝜑𝜒)
2 ex-natded5.2.2 . . 3 (𝜑 → (𝜒𝜓))
31, 2mpd 15 . 2 (𝜑𝜓)
4 ex-natded5.2.1 . 2 (𝜑 → ((𝜓𝜒) → 𝜃))
53, 1, 4mp2and 710 1 (𝜑𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 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 195  df-an 384
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator