Theorem tsna1 38853

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

Assertion

Ref	Expression
tsna1	⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓)))

Proof of Theorem tsna1

Step	Hyp	Ref	Expression
1		tsan1 38850	. 2 ⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
2		notnotb 318	. . . . 5 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ ¬ (𝜑 ⊼ 𝜓))
3		df-nan 1522	. . . . 5 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
4	2, 3	bitr3i 280	. . . 4 ⊢ (¬ ¬ (𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
5	4	con4bii 324	. . 3 ⊢ (¬ (𝜑 ⊼ 𝜓) ↔ (𝜑 ∧ 𝜓))
6	5	orbi2i 926	. 2 ⊢ (((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓)) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
7	1, 6	sylibr 237	1 ⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 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)