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Theorem tsbi3 33112
 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 209 . . . . 5 ((𝜑𝜓) → (𝜓𝜑))
2 con34b 305 . . . . . 6 ((𝜓𝜑) ↔ (¬ 𝜑 → ¬ 𝜓))
3 pm2.54 388 . . . . . 6 ((¬ 𝜑 → ¬ 𝜓) → (𝜑 ∨ ¬ 𝜓))
42, 3sylbi 206 . . . . 5 ((𝜓𝜑) → (𝜑 ∨ ¬ 𝜓))
51, 4syl 17 . . . 4 ((𝜑𝜓) → (𝜑 ∨ ¬ 𝜓))
65con3i 149 . . 3 (¬ (𝜑 ∨ ¬ 𝜓) → ¬ (𝜑𝜓))
76orri 390 . 2 ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓))
87a1i 11 1 (𝜃 → ((𝜑 ∨ ¬ 𝜓) ∨ ¬ (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:  tsbi4  33113  tsxo3  33116  mpt2bi123f  33141  mptbi12f  33145  ac6s6  33150
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