MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  orim12dALT Structured version   Visualization version   GIF version

Theorem orim12dALT 907
Description: Alternate proof of orim12d 960 which does not depend on df-an 397. This is an illustration of the conservativity of definitions (definitions do not permit to prove additional theorems whose statements do not contain the defined symbol). (Contributed by Wolf Lammen, 8-Aug-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
orim12dALT.1 (𝜑 → (𝜓𝜒))
orim12dALT.2 (𝜑 → (𝜃𝜏))
Assertion
Ref Expression
orim12dALT (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))

Proof of Theorem orim12dALT
StepHypRef Expression
1 pm2.53 847 . 2 ((𝜓𝜃) → (¬ 𝜓𝜃))
2 orim12dALT.1 . . . 4 (𝜑 → (𝜓𝜒))
32con3d 155 . . 3 (𝜑 → (¬ 𝜒 → ¬ 𝜓))
4 orim12dALT.2 . . 3 (𝜑 → (𝜃𝜏))
53, 4imim12d 81 . 2 (𝜑 → ((¬ 𝜓𝜃) → (¬ 𝜒𝜏)))
6 pm2.54 848 . 2 ((¬ 𝜒𝜏) → (𝜒𝜏))
71, 5, 6syl56 36 1 (𝜑 → ((𝜓𝜃) → (𝜒𝜏)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 843
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 208  df-or 844
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator