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Theorem tsbi1 32909
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 pm5.1 897 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
21olcd 406 . . 3 ((𝜑𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
3 pm3.13 520 . . . 4 (¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))
43orcd 405 . . 3 (¬ (𝜑𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
52, 4pm2.61i 174 . 2 ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓))
65a1i 11 1 (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382
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 195  df-or 383  df-an 384
This theorem is referenced by:  tsxo1  32913  mpt2bi123f  32940  mptbi12f  32944  ac6s6  32949
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