Theorem nic-mp 1587

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 1589. 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 1443	. . . 4 ⊢ ((𝜑 ⊼ (𝜒 ⊼ 𝜓)) ↔ (𝜑 → (𝜒 ∧ 𝜓)))
4	2, 3	mpbi 219	. . 3 ⊢ (𝜑 → (𝜒 ∧ 𝜓))
5	4	simprd 478	. 2 ⊢ (𝜑 → 𝜓)
6	1, 5	ax-mp 5	1 ⊢ 𝜓

Syntax hints:   → wi 4   ∧ wa 383   ⊼ wnan 1439

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 196  df-an 385  df-nan 1440

This theorem is referenced by:  nic-imp  1591  nic-idlem2  1593  nic-id  1594  nic-swap  1595  nic-isw1  1596  nic-isw2  1597  nic-iimp1  1598  nic-idel  1600  nic-ich  1601  nic-stdmp  1606  nic-luk1  1607  nic-luk2  1608  nic-luk3  1609  lukshefth1  1611  lukshefth2  1612  renicax  1613