MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ex-natded5.3-2 Structured version   Visualization version   GIF version

Theorem ex-natded5.3-2 27602
Description: A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 27601 and ex-natded5.3i 27603. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ex-natded5.3.1 (𝜑 → (𝜓𝜒))
ex-natded5.3.2 (𝜑 → (𝜒𝜃))
Assertion
Ref Expression
ex-natded5.3-2 (𝜑 → (𝜓 → (𝜒𝜃)))

Proof of Theorem ex-natded5.3-2
StepHypRef Expression
1 ex-natded5.3.1 . 2 (𝜑 → (𝜓𝜒))
2 ex-natded5.3.2 . . 3 (𝜑 → (𝜒𝜃))
31, 2syld 47 . 2 (𝜑 → (𝜓𝜃))
41, 3jcad 504 1 (𝜑 → (𝜓 → (𝜒𝜃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384
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 198  df-an 385
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator