MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl6d Structured version   Unicode version

Theorem syl6d 67
Description: A nested syllogism deduction. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Josh Purinton, 29-Dec-2000.) (Proof shortened by O'Cat, 2-Feb-2006.)
Hypotheses
Ref Expression
syl6d.1  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
syl6d.2  |-  ( ph  ->  ( th  ->  ta ) )
Assertion
Ref Expression
syl6d  |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )

Proof of Theorem syl6d
StepHypRef Expression
1 syl6d.1 . 2  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
2 syl6d.2 . . 3  |-  ( ph  ->  ( th  ->  ta ) )
32a1d 24 . 2  |-  ( ph  ->  ( ps  ->  ( th  ->  ta ) ) )
41, 3syldd 64 1  |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4
This theorem is referenced by:  syl8  68  cbv1hOLD  1978  sbi1  2136  omlimcl  6850  ltexprlem7  8950  axpre-sup  9075  fzm1  11158  caubnd  12193  ubthlem1  22403  ee13  28684  ssralv2  28713  rspsbc2  28716  truniALT  28724  cbv1hwAUX7  29609  sbi1NEW7  29661  cbv1hOLD7  29817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7
  Copyright terms: Public domain W3C validator