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Theorem tsbi4 33113
 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 33112 . 2 (𝜃 → ((𝜓 ∨ ¬ 𝜑) ∨ ¬ (𝜓𝜑)))
2 orcom 401 . . 3 ((𝜓 ∨ ¬ 𝜑) ↔ (¬ 𝜑𝜓))
3 bicom 211 . . . 4 ((𝜓𝜑) ↔ (𝜑𝜓))
43notbii 309 . . 3 (¬ (𝜓𝜑) ↔ ¬ (𝜑𝜓))
52, 4orbi12i 542 . 2 (((𝜓 ∨ ¬ 𝜑) ∨ ¬ (𝜓𝜑)) ↔ ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))
61, 5sylib 207 1 (𝜃 → ((¬ 𝜑𝜓) ∨ ¬ (𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ 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 196  df-or 384 This theorem is referenced by:  tsxo4  33117  mpt2bi123f  33141  mptbi12f  33145  ac6s6  33150
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