Theorem tsna1 36229

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 36226	. 2 ⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
2		notnotb 314	. . . . 5 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ ¬ (𝜑 ⊼ 𝜓))
3		df-nan 1484	. . . . 5 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
4	2, 3	bitr3i 276	. . . 4 ⊢ (¬ ¬ (𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
5	4	con4bii 320	. . 3 ⊢ (¬ (𝜑 ⊼ 𝜓) ↔ (𝜑 ∧ 𝜓))
6	5	orbi2i 909	. 2 ⊢ (((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓)) ↔ ((¬ 𝜑 ∨ ¬ 𝜓) ∨ (𝜑 ∧ 𝜓)))
7	1, 6	sylibr 233	1 ⊢ (𝜃 → ((¬ 𝜑 ∨ ¬ 𝜓) ∨ ¬ (𝜑 ⊼ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   ⊼ wnan 1483

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 206  df-an 396  df-or 844  df-nan 1484

This theorem is referenced by: (None)