Theorem nic-swap 1444

Description: ⊼ 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 1443	. 2 ⊢ (φ ⊼ (φ ⊼ φ))
2		nic-ax 1438	. 2 ⊢ ((φ ⊼ (φ ⊼ φ)) ⊼ ((τ ⊼ (τ ⊼ τ)) ⊼ ((θ ⊼ φ) ⊼ ((φ ⊼ θ) ⊼ (φ ⊼ θ)))))
3	1, 2	nic-mp 1436	1 ⊢ ((θ ⊼ φ) ⊼ ((φ ⊼ θ) ⊼ (φ ⊼ θ)))

Syntax hints:   ⊼ wnan 1287

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 177  df-an 360  df-nan 1288

This theorem is referenced by:  nic-isw1  1445  nic-isw2  1446  nic-bijust  1452  nic-luk1  1456