Theorem nic-mp 1673

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 1675. 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 1499	. . . 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 1493

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 1494

This theorem is referenced by:  nic-imp  1677  nic-idlem2  1679  nic-id  1680  nic-swap  1681  nic-isw1  1682  nic-isw2  1683  nic-iimp1  1684  nic-idel  1686  nic-ich  1687  nic-stdmp  1692  nic-luk1  1693  nic-luk2  1694  nic-luk3  1695  lukshefth1  1697  lukshefth2  1698  renicax  1699