Theorem nic-mpALT 1437

Description: A direct proof of nic-mp 1436. (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 1288	. . . . . 6 ⊢ ((φ ⊼ (χ ⊼ ψ)) ↔ ¬ (φ ∧ (χ ⊼ ψ)))
4		df-nan 1288	. . . . . . 7 ⊢ ((χ ⊼ ψ) ↔ ¬ (χ ∧ ψ))
5	4	anbi2i 675	. . . . . 6 ⊢ ((φ ∧ (χ ⊼ ψ)) ↔ (φ ∧ ¬ (χ ∧ ψ)))
6	3, 5	xchbinx 301	. . . . 5 ⊢ ((φ ⊼ (χ ⊼ ψ)) ↔ ¬ (φ ∧ ¬ (χ ∧ ψ)))
7	2, 6	mpbi 199	. . . 4 ⊢ ¬ (φ ∧ ¬ (χ ∧ ψ))
8		iman 413	. . . 4 ⊢ ((φ → (χ ∧ ψ)) ↔ ¬ (φ ∧ ¬ (χ ∧ ψ)))
9	7, 8	mpbir 200	. . 3 ⊢ (φ → (χ ∧ ψ))
10	9	simprd 449	. 2 ⊢ (φ → ψ)
11	1, 10	ax-mp 5	1 ⊢ ψ

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ⊼ wnan 1287

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 177  df-an 360  df-nan 1288

This theorem is referenced by: (None)