Theorem nic-swap 1712

Description: The connector ⊼ is symmetric. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nic-swap	⊢ ((𝜃 ⊼ 𝜑) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))

Proof of Theorem nic-swap

Step	Hyp	Ref	Expression
1		nic-id 1711	. 2 ⊢ (𝜑 ⊼ (𝜑 ⊼ 𝜑))
2		nic-ax 1706	. 2 ⊢ ((𝜑 ⊼ (𝜑 ⊼ 𝜑)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜑) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
3	1, 2	nic-mp 1704	1 ⊢ ((𝜃 ⊼ 𝜑) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))

This proof depends on syntax axioms:   ⊼ 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-nan 1522

This theorem is used by:  nic-isw1  1713  nic-isw2  1714  nic-bijust  1720  nic-luk1  1724