Theorem nic-mp 1671

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 1673. 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 1497	. . . 4 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ↔ (𝜑 → (𝜒 ∧ 𝜓)))
4	2, 3	mpbi 230	. . 3 ⊢ (𝜑 → (𝜒 ∧ 𝜓))
5	4	simprd 495	. 2 ⊢ (𝜑 → 𝜓)
6	1, 5	ax-mp 5	1 ⊢ 𝜓

Syntax hints:   → wi 4   ∧ wa 395   ⊼ wnan 1491

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 207  df-an 396  df-nan 1492

This theorem is referenced by:  nic-imp  1675  nic-idlem2  1677  nic-id  1678  nic-swap  1679  nic-isw1  1680  nic-isw2  1681  nic-iimp1  1682  nic-idel  1684  nic-ich  1685  nic-stdmp  1690  nic-luk1  1691  nic-luk2  1692  nic-luk3  1693  lukshefth1  1695  lukshefth2  1696  renicax  1697