Theorem nic-isw2 1689

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 1687	. . . 4 ⊢ ((𝜑 ⊼ 𝜃) ⊼ ((𝜃 ⊼ 𝜑) ⊼ (𝜃 ⊼ 𝜑)))
3	2	nic-imp 1683	. . 3 ⊢ ((𝜓 ⊼ (𝜃 ⊼ 𝜑)) ⊼ (((𝜑 ⊼ 𝜃) ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜃) ⊼ 𝜓)))
4	1, 3	nic-mp 1679	. 2 ⊢ ((𝜑 ⊼ 𝜃) ⊼ 𝜓)
5	4	nic-isw1 1688	1 ⊢ (𝜓 ⊼ (𝜑 ⊼ 𝜃))

Syntax hints:   ⊼ wnan 1487

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 210  df-an 400  df-nan 1488

This theorem is referenced by:  nic-bi1  1696  nic-bi2  1697  nic-luk1  1699  nic-luk2  1700