ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl121anc GIF version

Theorem syl121anc 1179
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)
Hypotheses
Ref Expression
sylXanc.1 (𝜑𝜓)
sylXanc.2 (𝜑𝜒)
sylXanc.3 (𝜑𝜃)
sylXanc.4 (𝜑𝜏)
syl121anc.5 ((𝜓 ∧ (𝜒𝜃) ∧ 𝜏) → 𝜂)
Assertion
Ref Expression
syl121anc (𝜑𝜂)

Proof of Theorem syl121anc
StepHypRef Expression
1 sylXanc.1 . 2 (𝜑𝜓)
2 sylXanc.2 . . 3 (𝜑𝜒)
3 sylXanc.3 . . 3 (𝜑𝜃)
42, 3jca 300 . 2 (𝜑 → (𝜒𝜃))
5 sylXanc.4 . 2 (𝜑𝜏)
6 syl121anc.5 . 2 ((𝜓 ∧ (𝜒𝜃) ∧ 𝜏) → 𝜂)
71, 4, 5, 6syl3anc 1174 1 (𝜑𝜂)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-3an 926
This theorem is referenced by:  syl122anc  1183  tfisi  4402  tfrcllemsucfn  6118  sbthlemi6  6669  sbthlemi8  6671  div32apd  8279  div13apd  8280  expdivapd  10096  modfsummodlemstep  10847
  Copyright terms: Public domain W3C validator