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Theorem tsbi1 34272
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 pm5.1 821 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
21olcd 863 . . 3 ((𝜑𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
3 pm3.13 979 . . . 4 (¬ (𝜑𝜓) → (¬ 𝜑 ∨ ¬ 𝜓))
43orcd 862 . . 3 (¬ (𝜑𝜓) → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
52, 4pm2.61i 176 . 2 ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓))
65a1i 11 1 (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 836
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-an 383  df-or 837
This theorem is referenced by:  tsxo1  34276  mpt2bi123f  34303  mptbi12f  34307  ac6s6  34312
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