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

Theorem sylnib 601
Description: A mixed syllogism inference from an implication and a biconditional. (Contributed by Wolf Lammen, 16-Dec-2013.)
Hypotheses
Ref Expression
sylnib.1 (𝜑 → ¬ 𝜓)
sylnib.2 (𝜓𝜒)
Assertion
Ref Expression
sylnib (𝜑 → ¬ 𝜒)

Proof of Theorem sylnib
StepHypRef Expression
1 sylnib.1 . 2 (𝜑 → ¬ 𝜓)
2 sylnib.2 . . 3 (𝜓𝜒)
32a1i 9 . 2 (𝜑 → (𝜓𝜒))
41, 3mtbid 597 1 (𝜑 → ¬ 𝜒)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  sylnibr  602  ssnelpss  3289  inssdif0im  3291  ordtriexmidlem2  4246  dmsn0el  4790  fidifsnen  6331  ltpopr  6691  caucvgprprlemnbj  6789  xrlttri3  8716  fzneuz  8961
  Copyright terms: Public domain W3C validator