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

Theorem annimim 775
Description: Express conjunction in terms of implication. One direction of Theorem *4.61 of [WhiteheadRussell] p. 120. The converse holds for decidable propositions, as can be seen at annimdc 833. (Contributed by Jim Kingdon, 24-Dec-2017.)
Assertion
Ref Expression
annimim ((φ ¬ ψ) → ¬ (φψ))

Proof of Theorem annimim
StepHypRef Expression
1 pm2.27 35 . . 3 (φ → ((φψ) → ψ))
2 con3 558 . . 3 (((φψ) → ψ) → (¬ ψ → ¬ (φψ)))
31, 2syl 14 . 2 (φ → (¬ ψ → ¬ (φψ)))
43imp 115 1 ((φ ¬ ψ) → ¬ (φψ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-in1 532  ax-in2 533
This theorem is referenced by:  pm4.65r  776  dcim  777  imanim  778  pm4.52im  796  exanaliim  1521
  Copyright terms: Public domain W3C validator