Theorem nic-id 1710

Description: Theorem id 23 expressed with ⊼. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nic-id	⊢ (𝜏 ⊼ (𝜏 ⊼ 𝜏))

Proof of Theorem nic-id

Step	Hyp	Ref	Expression
1		nic-ax 1705	. . 3 ⊢ ((𝜓 ⊼ (𝜓 ⊼ 𝜓)) ⊼ ((𝜃 ⊼ (𝜃 ⊼ 𝜃)) ⊼ ((𝜑 ⊼ 𝜓) ⊼ ((𝜓 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜑)))))
2	1	nic-idlem2 1709	. 2 ⊢ ((((𝜑 ⊼ 𝜓) ⊼ ((𝜓 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜑))) ⊼ (𝜒 ⊼ (𝜒 ⊼ 𝜒))) ⊼ (𝜓 ⊼ (𝜓 ⊼ 𝜓)))
3		nic-idlem1 1708	. . 3 ⊢ (((𝜒 ⊼ (𝜒 ⊼ 𝜒)) ⊼ (𝜏 ⊼ (𝜏 ⊼ 𝜏))) ⊼ ((((𝜑 ⊼ 𝜓) ⊼ ((𝜓 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜑))) ⊼ (𝜒 ⊼ (𝜒 ⊼ 𝜒))) ⊼ (((𝜑 ⊼ 𝜓) ⊼ ((𝜓 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜑))) ⊼ (𝜒 ⊼ (𝜒 ⊼ 𝜒)))))
4	3	nic-idlem2 1709	. 2 ⊢ (((((𝜑 ⊼ 𝜓) ⊼ ((𝜓 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜑))) ⊼ (𝜒 ⊼ (𝜒 ⊼ 𝜒))) ⊼ (𝜓 ⊼ (𝜓 ⊼ 𝜓))) ⊼ ((𝜒 ⊼ (𝜒 ⊼ 𝜒)) ⊼ (𝜏 ⊼ (𝜏 ⊼ 𝜏))))
5	2, 4	nic-mp 1703	1 ⊢ (𝜏 ⊼ (𝜏 ⊼ 𝜏))

This proof depends on syntax axioms:   ⊼ wnan 1520

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 1521

This theorem is used by:  nic-swap  1711  nic-idel  1716  nic-bi1  1720  nic-bi2  1721  nic-luk2  1724  nic-luk3  1725