Theorem nic-iimp1 1685

Description: Inference version of nic-imp 1678 using right-handed term. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nic-iimp1.1	⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓))
nic-iimp1.2	⊢ (𝜃 ⊼ 𝜒)

Assertion

Ref	Expression
nic-iimp1	⊢ (𝜃 ⊼ 𝜑)

Proof of Theorem nic-iimp1

Step	Hyp	Ref	Expression
1		nic-iimp1.2	. . 3 ⊢ (𝜃 ⊼ 𝜒)
2		nic-iimp1.1	. . . 4 ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓))
3	2	nic-imp 1678	. . 3 ⊢ ((𝜃 ⊼ 𝜒) ⊼ ((𝜑 ⊼ 𝜃) ⊼ (𝜑 ⊼ 𝜃)))
4	1, 3	nic-mp 1674	. 2 ⊢ (𝜑 ⊼ 𝜃)
5	4	nic-isw1 1683	1 ⊢ (𝜃 ⊼ 𝜑)

Syntax hints:   ⊼ wnan 1486

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 206  df-an 397  df-nan 1487

This theorem is referenced by:  nic-iimp2  1686  nic-bi1  1691  nic-bi2  1692  nic-luk2  1695  nic-luk3  1696