Users' Mathboxes Mathbox for Giovanni Mascellani < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tsbi4 Structured version   Visualization version   GIF version

Theorem tsbi4 35567
Description: A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
Assertion
Ref Expression
tsbi4 (𝜃 → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))

Proof of Theorem tsbi4
StepHypRef Expression
1 tsbi3 35566 . 2 (𝜃 → ((𝜓 ∨ ¬ 𝜑) ∨ ¬ (𝜓𝜑)))
2 orcom 867 . . 3 ((𝜓 ∨ ¬ 𝜑) ↔ (¬ 𝜑𝜓))
3 bicom 225 . . . 4 ((𝜓𝜑) ↔ (𝜑𝜓))
43notbii 323 . . 3 (¬ (𝜓𝜑) ↔ ¬ (𝜑𝜓))
52, 4orbi12i 912 . 2 (((𝜓 ∨ ¬ 𝜑) ∨ ¬ (𝜓𝜑)) ↔ ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))
61, 5sylib 221 1 (𝜃 → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wo 844
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 210  df-or 845
This theorem is referenced by:  tsxo4  35571  mpobi123f  35593  mptbi12f  35597  ac6s6  35603
  Copyright terms: Public domain W3C validator