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

Theorem syland 603
Description: A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
Hypotheses
Ref Expression
syland.1 (𝜑 → (𝜓𝜒))
syland.2 (𝜑 → ((𝜒𝜃) → 𝜏))
Assertion
Ref Expression
syland (𝜑 → ((𝜓𝜃) → 𝜏))

Proof of Theorem syland
StepHypRef Expression
1 syland.1 . . 3 (𝜑 → (𝜓𝜒))
2 syland.2 . . . 4 (𝜑 → ((𝜒𝜃) → 𝜏))
32expd 416 . . 3 (𝜑 → (𝜒 → (𝜃𝜏)))
41, 3syld 47 . 2 (𝜑 → (𝜓 → (𝜃𝜏)))
54impd 411 1 (𝜑 → ((𝜓𝜃) → 𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396
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  df-an 397
This theorem is referenced by:  sylani  604  sylan2d  605  syl2and  608  onfununi  8343  lt2add  11703  nn0seqcvgd  16511  1stcelcls  23185  llyidm  23212  filuni  23609  ballotlemimin  33790  btwnintr  35283  ifscgr  35308  btwnconn1lem12  35362  poimir  36824  cvrntr  38599  goldbachthlem2  46513
  Copyright terms: Public domain W3C validator