Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  3jcad Structured version   Visualization version   GIF version

 Description: Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)
Hypotheses
Ref Expression
Assertion
Ref Expression
3jcad (𝜑 → (𝜓 → (𝜒𝜃𝜏)))

StepHypRef Expression
1 3jcad.1 . . . 4 (𝜑 → (𝜓𝜒))
21imp 444 . . 3 ((𝜑𝜓) → 𝜒)
3 3jcad.2 . . . 4 (𝜑 → (𝜓𝜃))
43imp 444 . . 3 ((𝜑𝜓) → 𝜃)
5 3jcad.3 . . . 4 (𝜑 → (𝜓𝜏))
65imp 444 . . 3 ((𝜑𝜓) → 𝜏)
72, 4, 63jca 1235 . 2 ((𝜑𝜓) → (𝜒𝜃𝜏))
87ex 449 1 (𝜑 → (𝜓 → (𝜒𝜃𝜏)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031 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-3an 1033 This theorem is referenced by:  onfununi  7325  uzm1  11594  ixxssixx  12060  iccid  12091  iccsplit  12176  fzen  12229  lmodprop2d  18748  fbun  21454  hausflim  21595  icoopnst  22546  iocopnst  22547  abelth  23999  eupai  26494  shsvs  27566  cnlnssadj  28323  cvmlift2lem10  30548  endofsegid  31362  elicc3  31481  areacirclem1  32670  islvol2aN  33896  alrim3con13v  37764  bgoldbtbndlem4  40224  usgr2pth  40970
 Copyright terms: Public domain W3C validator