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Theorem tsbi3 35281
Description: A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
Assertion
Ref Expression
tsbi3 (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))

Proof of Theorem tsbi3
StepHypRef Expression
1 biimpr 221 . . . . 5 ((𝜑𝜓) → (𝜓𝜑))
2 con34b 317 . . . . . 6 ((𝜓𝜑) ↔ (¬ 𝜑 → ¬ 𝜓))
3 pm2.54 848 . . . . . 6 ((¬ 𝜑 → ¬ 𝜓) → (𝜑 ∨ ¬ 𝜓))
42, 3sylbi 218 . . . . 5 ((𝜓𝜑) → (𝜑 ∨ ¬ 𝜓))
51, 4syl 17 . . . 4 ((𝜑𝜓) → (𝜑 ∨ ¬ 𝜓))
65con3i 157 . . 3 (¬ (𝜑 ∨ ¬ 𝜓) → ¬ (𝜑𝜓))
76orri 858 . 2 ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓))
87a1i 11 1 (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wo 843
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 208  df-or 844
This theorem is referenced by:  tsbi4  35282  tsxo3  35285  mpobi123f  35308  mptbi12f  35312  ac6s6  35318
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