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Theorem tsna1 36069
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 36066 . 2 (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
2 notnotb 318 . . . . 5 ((𝜑𝜓) ↔ ¬ ¬ (𝜑𝜓))
3 df-nan 1488 . . . . 5 ((𝜑𝜓) ↔ ¬ (𝜑𝜓))
42, 3bitr3i 280 . . . 4 (¬ ¬ (𝜑𝜓) ↔ ¬ (𝜑𝜓))
54con4bii 324 . . 3 (¬ (𝜑𝜓) ↔ (𝜑𝜓))
65orbi2i 913 . 2 (((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑𝜓)))
71, 6sylibr 237 1 (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 847  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 210  df-an 400  df-or 848  df-nan 1488
This theorem is referenced by: (None)
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