MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imp42 Structured version   Visualization version   GIF version

Theorem imp42 427
Description: An importation inference. (Contributed by NM, 26-Apr-1994.)
Hypothesis
Ref Expression
imp4.1 (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
Assertion
Ref Expression
imp42 (((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)

Proof of Theorem imp42
StepHypRef Expression
1 imp4.1 . . 3 (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
21imp32 419 . 2 ((𝜑 ∧ (𝜓𝜒)) → (𝜃𝜏))
32imp 407 1 (((𝜑 ∧ (𝜓𝜒)) ∧ 𝜃) → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396
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 206  df-an 397
This theorem is referenced by:  imp55  443  ltexprlem7  11033  iscatd  17613  isposd  18272  pospropd  18276  mulgghm2  21037  ordtbaslem  22683  txbas  23062  nocvxminlem  27268  frgrncvvdeqlem8  29548  grporcan  29758  chirredlem1  31630  cvxpconn  34221  cvxsconn  34222  rngonegmn1l  36797  prnc  36923  reuopreuprim  46180
  Copyright terms: Public domain W3C validator