Theorem nic-stdmp 1455

Description: Derive the standard modus ponens from nic-mp 1436. (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 1434	. . . 4 ⊢ (((φ ⊼ (ψ ⊼ ψ)) ⊼ (φ → ψ)) ⊼ (((φ ⊼ (ψ ⊼ ψ)) ⊼ (φ ⊼ (ψ ⊼ ψ))) ⊼ ((φ → ψ) ⊼ (φ → ψ))))
4	3	nic-bi2 1454	. . 3 ⊢ ((φ → ψ) ⊼ ((φ ⊼ (ψ ⊼ ψ)) ⊼ (φ ⊼ (ψ ⊼ ψ))))
5	2, 4	nic-mp 1436	. 2 ⊢ (φ ⊼ (ψ ⊼ ψ))
6	1, 5	nic-mp 1436	1 ⊢ ψ

This proof depends on syntax axioms:   → wi 4   ⊼ wnan 1287

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288

This theorem is used by: (None)