Theorem nic-stdmp 1723

Description: Derive the standard modus ponens from nic-mp 1704. (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 1702	. . . 4 ⊢ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 → 𝜓)) ⊼ (((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜓))) ⊼ ((𝜑 → 𝜓) ⊼ (𝜑 → 𝜓))))
4	3	nic-bi2 1722	. . 3 ⊢ ((𝜑 → 𝜓) ⊼ ((𝜑 ⊼ (𝜓 ⊼ 𝜓)) ⊼ (𝜑 ⊼ (𝜓 ⊼ 𝜓))))
5	2, 4	nic-mp 1704	. 2 ⊢ (𝜑 ⊼ (𝜓 ⊼ 𝜓))
6	1, 5	nic-mp 1704	1 ⊢ 𝜓

This proof depends on syntax axioms:   → wi 4   ⊼ 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-or 862  df-nan 1522

This theorem is used by: (None)