Users' Mathboxes Mathbox for Giovanni Mascellani < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tsbi1 Structured version   Visualization version   GIF version

Theorem tsbi1 33110
Description: A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
Assertion
Ref Expression
tsbi1 (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))

Proof of Theorem tsbi1
StepHypRef Expression
1 pm5.1 898 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
21olcd 407 . . 3 ((𝜑𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
3 pm3.13 521 . . . 4 (¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))
43orcd 406 . . 3 (¬ (𝜑𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
52, 4pm2.61i 175 . 2 ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓))
65a1i 11 1 (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383
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 196  df-or 384  df-an 385
This theorem is referenced by:  tsxo1  33114  mpt2bi123f  33141  mptbi12f  33145  ac6s6  33150
  Copyright terms: Public domain W3C validator