Users' Mathboxes Mathbox for Giovanni Mascellani < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ts3or1 Structured version   Visualization version   GIF version

Theorem ts3or1 33631
Description: A Tseitin axiom for triple logical disjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
Assertion
Ref Expression
ts3or1 (𝜃 → (((𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑𝜓𝜒)))

Proof of Theorem ts3or1
StepHypRef Expression
1 exmidd 432 . 2 (𝜃 → (((𝜑𝜓) ∨ 𝜒) ∨ ¬ ((𝜑𝜓) ∨ 𝜒)))
2 df-3or 1037 . . . 4 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∨ 𝜒))
32notbii 310 . . 3 (¬ (𝜑𝜓𝜒) ↔ ¬ ((𝜑𝜓) ∨ 𝜒))
43orbi2i 541 . 2 ((((𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑𝜓𝜒)) ↔ (((𝜑𝜓) ∨ 𝜒) ∨ ¬ ((𝜑𝜓) ∨ 𝜒)))
51, 4sylibr 224 1 (𝜃 → (((𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  w3o 1035
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 385  df-3or 1037
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator