Theorem nic-imp 1708

Description: Inference for nic-mp 1704 using nic-ax 1706 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 1706	. 2 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ⊼ ((𝜏 ⊼ (𝜏 ⊼ 𝜏)) ⊼ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))))
3	1, 2	nic-mp 1704	1 ⊢ ((𝜃 ⊼ 𝜒) ⊼ ((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-idlem1  1709  nic-idlem2  1710  nic-isw2  1714  nic-iimp1  1715  nic-idel  1717  nic-ich  1718  nic-idbl  1719  nic-luk1  1724