MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpii Structured version   Visualization version   GIF version

Theorem mpii 46
Description: A doubly nested modus ponens inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.)
Hypotheses
Ref Expression
mpii.1 𝜒
mpii.2 (𝜑 → (𝜓 → (𝜒𝜃)))
Assertion
Ref Expression
mpii (𝜑 → (𝜓𝜃))

Proof of Theorem mpii
StepHypRef Expression
1 mpii.1 . . 3 𝜒
21a1i 11 . 2 (𝜓𝜒)
3 mpii.2 . 2 (𝜑 → (𝜓 → (𝜒𝜃)))
42, 3mpdi 45 1 (𝜑 → (𝜓𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7
This theorem is referenced by:  intmin  4887  dfiin2g  4948  ssorduni  7489  suceloni  7517  lublecllem  17586  irredmul  19388  opnneiid  21662  isufil2  22444  mdbr3  30001  mdbr4  30002  dmdbr5  30012  filnetlem4  33626  iunord  44707
  Copyright terms: Public domain W3C validator