MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  impcon4bid Structured version   Visualization version   GIF version

Theorem impcon4bid 217
Description: A variation on impbid 202 with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009.)
Hypotheses
Ref Expression
impcon4bid.1 (𝜑 → (𝜓𝜒))
impcon4bid.2 (𝜑 → (¬ 𝜓 → ¬ 𝜒))
Assertion
Ref Expression
impcon4bid (𝜑 → (𝜓𝜒))

Proof of Theorem impcon4bid
StepHypRef Expression
1 impcon4bid.1 . 2 (𝜑 → (𝜓𝜒))
2 impcon4bid.2 . . 3 (𝜑 → (¬ 𝜓 → ¬ 𝜒))
32con4d 114 . 2 (𝜑 → (𝜒𝜓))
41, 3impbid 202 1 (𝜑 → (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196
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 197
This theorem is referenced by:  con4bid  306  soisoi  6618  isomin  6627  alephdom  8942  nn0n0n1ge2b  11397  om2uzlt2i  12790  sadcaddlem  15226  isprm5  15466  pcdvdsb  15620  cvgdvgrat  38829
  Copyright terms: Public domain W3C validator