ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  syl23anc GIF version

Theorem syl23anc 1181
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)
sylXanc.5 (𝜑𝜂)
syl23anc.6 (((𝜓𝜒) ∧ (𝜃𝜏𝜂)) → 𝜁)
Assertion
Ref Expression
syl23anc (𝜑𝜁)

Proof of Theorem syl23anc
StepHypRef Expression
1 sylXanc.1 . . 3 (𝜑𝜓)
2 sylXanc.2 . . 3 (𝜑𝜒)
31, 2jca 300 . 2 (𝜑 → (𝜓𝜒))
4 sylXanc.3 . 2 (𝜑𝜃)
5 sylXanc.4 . 2 (𝜑𝜏)
6 sylXanc.5 . 2 (𝜑𝜂)
7 syl23anc.6 . 2 (((𝜓𝜒) ∧ (𝜃𝜏𝜂)) → 𝜁)
83, 4, 5, 6, 7syl13anc 1176 1 (𝜑𝜁)
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:  div2subapd  8275  gcdaddm  11068
  Copyright terms: Public domain W3C validator