MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl2an3an Structured version   Visualization version   GIF version

Theorem syl2an3an 1531
Description: syl3an 1162 with antecedents in standard conjunction form. (Contributed by Alan Sare, 31-Aug-2016.)
Hypotheses
Ref Expression
syl2an3an.1 (𝜑𝜓)
syl2an3an.2 (𝜑𝜒)
syl2an3an.3 (𝜃𝜏)
syl2an3an.4 ((𝜓𝜒𝜏) → 𝜂)
Assertion
Ref Expression
syl2an3an ((𝜑𝜃) → 𝜂)

Proof of Theorem syl2an3an
StepHypRef Expression
1 syl2an3an.1 . . 3 (𝜑𝜓)
2 syl2an3an.2 . . 3 (𝜑𝜒)
3 syl2an3an.3 . . 3 (𝜃𝜏)
4 syl2an3an.4 . . 3 ((𝜓𝜒𝜏) → 𝜂)
51, 2, 3, 4syl3an 1162 . 2 ((𝜑𝜑𝜃) → 𝜂)
653anidm12 1528 1 ((𝜑𝜃) → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070
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  df-an 383  df-3an 1072
This theorem is referenced by:  syl2an23an  1532  disjxiun  4781  funcnvtp  6092  cncongr1  15587  gausslemma2dlem2  25312  eucrctshift  27420  numclwlk1lem1  27555  fmtnofac2lem  41998
  Copyright terms: Public domain W3C validator