Theorem tsna1 33121

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 33118	. 2 ⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
2		notnotb 303	. . . . 5 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ ¬ (𝜑 ⊼ 𝜓))
3		df-nan 1440	. . . . 5 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
4	2, 3	bitr3i 265	. . . 4 ⊢ (¬ ¬ (𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
5	4	con4bii 310	. . 3 ⊢ (¬ (𝜑 ⊼ 𝜓) ↔ (𝜑 ∧ 𝜓))
6	5	orbi2i 540	. 2 ⊢ (((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓)) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
7	1, 6	sylibr 223	1 ⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ⊼ wnan 1439

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 196  df-or 384  df-an 385  df-nan 1440

This theorem is referenced by: (None)