Theorem nic-ich 1684

Description: Chained inference. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
nic-ich	⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜒))

Proof of Theorem nic-ich

Step	Hyp	Ref	Expression
1		nic-ich.2	. . 3 ⊢ (𝜓 ⊼ (𝜒 ⊼ 𝜒))
2	1	nic-isw1 1679	. 2 ⊢ ((𝜒 ⊼ 𝜒) ⊼ 𝜓)
3		nic-ich.1	. . 3 ⊢ (𝜑 ⊼ (𝜓 ⊼ 𝜓))
4	3	nic-imp 1674	. 2 ⊢ (((𝜒 ⊼ 𝜒) ⊼ 𝜓) ⊼ ((𝜑 ⊼ (𝜒 ⊼ 𝜒)) ⊼ (𝜑 ⊼ (𝜒 ⊼ 𝜒))))
5	2, 4	nic-mp 1670	1 ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜒))

Syntax hints:   ⊼ wnan 1490

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 207  df-an 396  df-nan 1491

This theorem is referenced by:  nic-idbl  1685  nic-luk1  1690