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Theorem tsbi2 33570
Description: A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
Assertion
Ref Expression
tsbi2 (𝜃 → ((𝜑𝜓) ∨ (𝜑𝜓)))

Proof of Theorem tsbi2
StepHypRef Expression
1 pm5.21 902 . . . 4 ((¬ 𝜑 ∧ ¬ 𝜓) → (𝜑𝜓))
21olcd 408 . . 3 ((¬ 𝜑 ∧ ¬ 𝜓) → ((𝜑𝜓) ∨ (𝜑𝜓)))
3 pm4.57 518 . . . . 5 (¬ (¬ 𝜑 ∧ ¬ 𝜓) ↔ (𝜑𝜓))
43biimpi 206 . . . 4 (¬ (¬ 𝜑 ∧ ¬ 𝜓) → (𝜑𝜓))
54orcd 407 . . 3 (¬ (¬ 𝜑 ∧ ¬ 𝜓) → ((𝜑𝜓) ∨ (𝜑𝜓)))
62, 5pm2.61i 176 . 2 ((𝜑𝜓) ∨ (𝜑𝜓))
76a1i 11 1 (𝜃 → ((𝜑𝜓) ∨ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384
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 385  df-an 386
This theorem is referenced by:  tsxo2  33574  mpt2bi123f  33600  mptbi12f  33604  ac6s6  33609
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