Theorem nic-swap

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

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

Proof

Step	Hyp	Ref	Expression
1		nic-id	⊢ ( 𝜑 ⊼ ( 𝜑 ⊼ 𝜑 ) )
2		nic-ax	⊢ ( ( 𝜑 ⊼ ( 𝜑 ⊼ 𝜑 ) ) ⊼ ( ( 𝜏 ⊼ ( 𝜏 ⊼ 𝜏 ) ) ⊼ ( ( 𝜃 ⊼ 𝜑 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( 𝜑 ⊼ 𝜃 ) ) ) ) )
3	1 2	nic-mp	⊢ ( ( 𝜃 ⊼ 𝜑 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( 𝜑 ⊼ 𝜃 ) ) )