Theorem nic-mpALT 1588

Description: A direct proof of nic-mp 1587. (Contributed by NM, 30-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nic-jmin	⊢ 𝜑
nic-jmaj	⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓))

Assertion

Ref	Expression
nic-mpALT	⊢ 𝜓

Proof of Theorem nic-mpALT

Step	Hyp	Ref	Expression
1		nic-jmin	. 2 ⊢ 𝜑
2		nic-jmaj	. . . . 5 ⊢ (𝜑 ⊼ (𝜒 ⊼ 𝜓))
3		df-nan 1440	. . . . . 6 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ↔ ¬ (𝜑 ∧ (𝜒 ⊼ 𝜓)))
4		df-nan 1440	. . . . . . 7 ⊢ ((𝜒 ⊼ 𝜓) ↔ ¬ (𝜒 ∧ 𝜓))
5	4	anbi2i 726	. . . . . 6 ⊢ ((𝜑 ∧ (𝜒 ⊼ 𝜓)) ↔ (𝜑 ∧ ¬ (𝜒 ∧ 𝜓)))
6	3, 5	xchbinx 323	. . . . 5 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ↔ ¬ (𝜑 ∧ ¬ (𝜒 ∧ 𝜓)))
7	2, 6	mpbi 219	. . . 4 ⊢ ¬ (𝜑 ∧ ¬ (𝜒 ∧ 𝜓))
8		iman 439	. . . 4 ⊢ ((𝜑 → (𝜒 ∧ 𝜓)) ↔ ¬ (𝜑 ∧ ¬ (𝜒 ∧ 𝜓)))
9	7, 8	mpbir 220	. . 3 ⊢ (𝜑 → (𝜒 ∧ 𝜓))
10	9	simprd 478	. 2 ⊢ (𝜑 → 𝜓)
11	1, 10	ax-mp 5	1 ⊢ 𝜓

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ⊼ wnan 1439

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 196  df-an 385  df-nan 1440

This theorem is referenced by: (None)