Theorem nic-mp 1436

Description: Derive Nicod's rule of modus ponens using 'nand', from the standard one. Although the major and minor premise together also imply χ, this form is necessary for useful derivations from nic-ax 1438. In a pure (standalone) treatment of Nicod's axiom, this theorem would be changed to an axiom ($a statement). (Contributed by Jeff Hoffman, 19-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
nic-mp	⊢ ψ

Proof of Theorem nic-mp

Step	Hyp	Ref	Expression
1		nic-jmin	. 2 ⊢ φ
2		nic-jmaj	. . . 4 ⊢ (φ ⊼ (χ ⊼ ψ))
3		nannan 1291	. . . 4 ⊢ ((φ ⊼ (χ ⊼ ψ)) ↔ (φ → (χ ∧ ψ)))
4	2, 3	mpbi 199	. . 3 ⊢ (φ → (χ ∧ ψ))
5	4	simprd 449	. 2 ⊢ (φ → ψ)
6	1, 5	ax-mp 5	1 ⊢ ψ

Syntax hints:   → 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:  nic-imp  1440  nic-idlem2  1442  nic-id  1443  nic-swap  1444  nic-isw1  1445  nic-isw2  1446  nic-iimp1  1447  nic-idel  1449  nic-ich  1450  nic-stdmp  1455  nic-luk1  1456  nic-luk2  1457  nic-luk3  1458  lukshefth1  1460  lukshefth2  1461  renicax  1462