MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl7bi Structured version   Visualization version   GIF version

Theorem syl7bi 254
Description: A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 14-May-1993.)
Hypotheses
Ref Expression
syl7bi.1 (𝜑𝜓)
syl7bi.2 (𝜒 → (𝜃 → (𝜓𝜏)))
Assertion
Ref Expression
syl7bi (𝜒 → (𝜃 → (𝜑𝜏)))

Proof of Theorem syl7bi
StepHypRef Expression
1 syl7bi.1 . . 3 (𝜑𝜓)
21biimpi 215 . 2 (𝜑𝜓)
3 syl7bi.2 . 2 (𝜒 → (𝜃 → (𝜓𝜏)))
42, 3syl7 74 1 (𝜒 → (𝜃 → (𝜑𝜏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205
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
This theorem is referenced by:  3jao  1425  rspct  3598  zfpair  5418  gruen  10803  axpre-sup  11160  nn0lt2  12621  fzofzim  13675  ndvdssub  16348  cyccom  19074  alexsubALT  23546  clwlkclwwlklem2a  29240  erclwwlktr  29264  erclwwlkntr  29313  fmlasuc  34365  dfon2lem8  34750  prtlem15  37733  prtlem18  37735  2reuimp0  45808
  Copyright terms: Public domain W3C validator