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Theorem tsna1 34081
 Description: A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
Assertion
Ref Expression
tsna1 (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))

Proof of Theorem tsna1
StepHypRef Expression
1 tsan1 34078 . 2 (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
2 notnotb 304 . . . . 5 ((𝜑𝜓) ↔ ¬ ¬ (𝜑𝜓))
3 df-nan 1488 . . . . 5 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
42, 3bitr3i 266 . . . 4 (¬ ¬ (𝜑𝜓) ↔ ¬ (𝜑𝜓))
54con4bii 310 . . 3 (¬ (𝜑𝜓) ↔ (𝜑𝜓))
65orbi2i 540 . 2 (((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
71, 6sylibr 224 1 (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ⊼ wnan 1487 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  df-an 385  df-nan 1488 This theorem is referenced by: (None)
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