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Theorem tsbi4 34274
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 34273 . 2 (𝜃 → ((𝜓 ∨ ¬ 𝜑) ∨ ¬ (𝜓𝜑)))
2 orcom 401 . . 3 ((𝜓 ∨ ¬ 𝜑) ↔ (¬ 𝜑𝜓))
3 bicom 212 . . . 4 ((𝜓𝜑) ↔ (𝜑𝜓))
43notbii 309 . . 3 (¬ (𝜓𝜑) ↔ ¬ (𝜑𝜓))
52, 4orbi12i 544 . 2 (((𝜓 ∨ ¬ 𝜑) ∨ ¬ (𝜓𝜑)) ↔ ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))
61, 5sylib 208 1 (𝜃 → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 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 197  df-or 384
This theorem is referenced by:  tsxo4  34278  mpt2bi123f  34302  mptbi12f  34306  ac6s6  34311
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