Theorem tsna3 33953

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 33950	. 2 ⊢ (𝜃 → (𝜓 ∨ ¬ (𝜑 ∧ 𝜓)))
2		df-nan 1448	. . 3 ⊢ ((𝜑 ⊼ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
3	2	orbi2i 541	. 2 ⊢ ((𝜓 ∨ (𝜑 ⊼ 𝜓)) ↔ (𝜓 ∨ ¬ (𝜑 ∧ 𝜓)))
4	1, 3	sylibr 224	1 ⊢ (𝜃 → (𝜓 ∨ (𝜑 ⊼ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   ⊼ wnan 1447

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 197  df-or 385  df-an 386  df-nan 1448

This theorem is referenced by: (None)