Theorem tsna3 39078

Description: A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)

Assertion

Ref	Expression
tsna3	⊢ (𝜃 → (𝜓 ∨ (𝜑 ⊼ 𝜓)))

Proof of Theorem tsna3

Step	Hyp	Ref	Expression
1		tsan3 39075	. 2 ⊢ (𝜃 → (𝜓 ∨ ¬ (𝜑 ∧ 𝜓)))
2		df-nan 1522	. . 3 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
3	2	orbi2i 926	. 2 ⊢ ((𝜓 ∨ (𝜑 ⊼ 𝜓)) ↔ (𝜓 ∨ ¬ (𝜑 ∧ 𝜓)))
4	1, 3	sylibr 237	1 ⊢ (𝜃 → (𝜓 ∨ (𝜑 ⊼ 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ⊼ wnan 1521

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-nan 1522

This theorem is used by: (None)