Theorem nic-imp 1698

Description: Inference for nic-mp 1694 using nic-ax 1696 as major premise. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nic-imp.1	⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓))

Assertion

Ref	Expression
nic-imp	⊢ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))

Proof of Theorem nic-imp

Step	Hyp	Ref	Expression
1		nic-imp.1	. 2 ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓))
2		nic-ax 1696	. 2 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
3	1, 2	nic-mp 1694	1 ⊢ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))

Syntax hints:   ⊼ wnan 1514

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 210  df-an 401  df-nan 1515

This theorem is referenced by:  nic-idlem1  1699  nic-idlem2  1700  nic-isw2  1704  nic-iimp1  1705  nic-idel  1707  nic-ich  1708  nic-idbl  1709  nic-luk1  1714