Theorem nic-isw2 1714

Description: Inference for swapping nested terms. (Contributed by Jeff Hoffman, 17-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nic-isw2.1	⊢ (𝜓 ⊼ (𝜃 ⊼ 𝜑))

Assertion

Ref	Expression
nic-isw2	⊢ (𝜓 ⊼ (𝜑 ⊼ 𝜃))

Proof of Theorem nic-isw2

Step	Hyp	Ref	Expression
1		nic-isw2.1	. . 3 ⊢ (𝜓 ⊼ (𝜃 ⊼ 𝜑))
2		nic-swap 1712	. . . 4 ⊢ ((𝜑 ⊼ 𝜃) ⊼ ((𝜃 ⊼ 𝜑) ⊼ (𝜃 ⊼ 𝜑)))
3	2	nic-imp 1708	. . 3 ⊢ ((𝜓 ⊼ (𝜃 ⊼ 𝜑)) ⊼ (((𝜑 ⊼ 𝜃) ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜃) ⊼ 𝜓)))
4	1, 3	nic-mp 1704	. 2 ⊢ ((𝜑 ⊼ 𝜃) ⊼ 𝜓)
5	4	nic-isw1 1713	1 ⊢ (𝜓 ⊼ (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-bi1  1721  nic-bi2  1722  nic-luk1  1724  nic-luk2  1725