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Theorem ts3an2 34088
Description: A Tseitin axiom for triple logical conjunction, in deduction form. (Contributed by Giovanni Mascellani, 25-Mar-2018.)
Assertion
Ref Expression
ts3an2 (𝜃 → ((𝜑𝜓) ∨ ¬ (𝜑𝜓𝜒)))

Proof of Theorem ts3an2
StepHypRef Expression
1 tsan2 34079 . 2 (𝜃 → ((𝜑𝜓) ∨ ¬ ((𝜑𝜓) ∧ 𝜒)))
2 df-3an 1056 . . . 4 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
32notbii 309 . . 3 (¬ (𝜑𝜓𝜒) ↔ ¬ ((𝜑𝜓) ∧ 𝜒))
43orbi2i 540 . 2 (((𝜑𝜓) ∨ ¬ (𝜑𝜓𝜒)) ↔ ((𝜑𝜓) ∨ ¬ ((𝜑𝜓) ∧ 𝜒)))
51, 4sylibr 224 1 (𝜃 → ((𝜑𝜓) ∨ ¬ (𝜑𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  w3a 1054
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-3an 1056
This theorem is referenced by: (None)
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