Theorem nic-stdmp 1698

Description: Derive the standard modus ponens from nic-mp 1679. (Contributed by Jeff Hoffman, 18-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nic-smin	⊢ 𝜑
nic-smaj	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
nic-stdmp	⊢ 𝜓

Proof of Theorem nic-stdmp

Step	Hyp	Ref	Expression
1		nic-smin	. 2 ⊢ 𝜑
2		nic-smaj	. . 3 ⊢ (𝜑 → 𝜓)
3		nic-dfim 1677	. . . 4 ⊢ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 → 𝜓)) ⊼ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜓))) ⊼ ((𝜑 → 𝜓) ⊼ (𝜑 → 𝜓))))
4	3	nic-bi2 1697	. . 3 ⊢ ((𝜑 → 𝜓) ⊼ ((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜓))))
5	2, 4	nic-mp 1679	. 2 ⊢ (𝜑 ⊼ (𝜓 ⊼ 𝜓))
6	1, 5	nic-mp 1679	1 ⊢ 𝜓

Syntax hints:   → wi 4   ⊼ 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-or 848  df-nan 1488

This theorem is referenced by: (None)