MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  syl6ci Structured version   Visualization version   GIF version

Theorem syl6ci 71
Description: A syllogism inference combined with contraction. (Contributed by Alan Sare, 18-Mar-2012.)
Hypotheses
Ref Expression
syl6ci.1 (𝜑 → (𝜓𝜒))
syl6ci.2 (𝜑𝜃)
syl6ci.3 (𝜒 → (𝜃𝜏))
Assertion
Ref Expression
syl6ci (𝜑 → (𝜓𝜏))

Proof of Theorem syl6ci
StepHypRef Expression
1 syl6ci.1 . 2 (𝜑 → (𝜓𝜒))
2 syl6ci.2 . . 3 (𝜑𝜃)
32a1d 25 . 2 (𝜑 → (𝜓𝜃))
4 syl6ci.3 . 2 (𝜒 → (𝜃𝜏))
51, 3, 4syl6c 70 1 (𝜑 → (𝜓𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7
This theorem is referenced by:  ordelord  5907  f1dmex  7303  omeulem2  7835  2pwuninel  8283  isumrpcl  14795  kqfvima  21756  caubl  23327  nbupgr  26461  nbumgrvtx  26463  umgr2adedgspth  27090  soseq  32082  btwnconn1lem12  32533  sbcim2g  39269  ee21an  39480
  Copyright terms: Public domain W3C validator