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Theorem tsbi1 35722
 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 pm5.1 822 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
21olcd 871 . . 3 ((𝜑𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
3 pm3.13 992 . . . 4 (¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))
43orcd 870 . . 3 (¬ (𝜑𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
52, 4pm2.61i 185 . 2 ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓))
65a1i 11 1 (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ 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-an 400  df-or 845 This theorem is referenced by:  tsxo1  35726  mpobi123f  35751  mptbi12f  35755  ac6s6  35761
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