Theorem nic-bi2 1722

Description: Inference to extract the other side of an implication from a 'biconditional' definition. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nic-bi2.1	⊢ ((𝜑 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓)))

Assertion

Ref	Expression
nic-bi2	⊢ (𝜓 ⊼ (𝜑 ⊼ 𝜑))

Proof of Theorem nic-bi2

Step	Hyp	Ref	Expression
1		nic-bi2.1	. . . 4 ⊢ ((𝜑 ⊼ 𝜓) ⊼ ((𝜑 ⊼ 𝜑) ⊼ (𝜓 ⊼ 𝜓)))
2	1	nic-isw2 1714	. . 3 ⊢ ((𝜑 ⊼ 𝜓) ⊼ ((𝜓 ⊼ 𝜓) ⊼ (𝜑 ⊼ 𝜑)))
3		nic-id 1711	. . 3 ⊢ (𝜓 ⊼ (𝜓 ⊼ 𝜓))
4	2, 3	nic-iimp1 1715	. 2 ⊢ (𝜓 ⊼ (𝜑 ⊼ 𝜓))
5	4	nic-idel 1717	1 ⊢ (𝜓 ⊼ (𝜑 ⊼ 𝜑))

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-stdmp  1723  nic-luk1  1724  nic-luk2  1725  nic-luk3  1726