Theorem nic-mp 1672

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 1674. 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 1498	. . . 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 1492

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 1493

This theorem is referenced by:  nic-imp  1676  nic-idlem2  1678  nic-id  1679  nic-swap  1680  nic-isw1  1681  nic-isw2  1682  nic-iimp1  1683  nic-idel  1685  nic-ich  1686  nic-stdmp  1691  nic-luk1  1692  nic-luk2  1693  nic-luk3  1694  lukshefth1  1696  lukshefth2  1697  renicax  1698