 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  exp45 Structured version   Visualization version   GIF version

Theorem exp45 640
 Description: An exportation inference. (Contributed by NM, 26-Apr-1994.)
Hypothesis
Ref Expression
exp45.1 ((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜏)
Assertion
Ref Expression
exp45 (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))

Proof of Theorem exp45
StepHypRef Expression
1 exp45.1 . . 3 ((𝜑 ∧ (𝜓 ∧ (𝜒𝜃))) → 𝜏)
21exp32 629 . 2 (𝜑 → (𝜓 → ((𝜒𝜃) → 𝜏)))
32exp4a 631 1 (𝜑 → (𝜓 → (𝜒 → (𝜃𝜏))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383 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 This theorem is referenced by:  oaass  7528  zorn2lem4  9204  zorn2lem7  9207  iscatd2  16165  fgss2  21488  alexsubALTlem4  21664  grporcan  26756  spansncvi  27895  mdsymlem5  28650  riotasv3d  33264  cvratlem  33725  hbtlem2  36713
 Copyright terms: Public domain W3C validator