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Theorem annimim 816
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 879. (Contributed by Jim Kingdon, 24-Dec-2017.)
Assertion
Ref Expression
annimim ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))

Proof of Theorem annimim
StepHypRef Expression
1 pm2.27 39 . . 3 (𝜑 → ((𝜑𝜓) → 𝜓))
2 con3 604 . . 3 (((𝜑𝜓) → 𝜓) → (¬ 𝜓 → ¬ (𝜑𝜓)))
31, 2syl 14 . 2 (𝜑 → (¬ 𝜓 → ¬ (𝜑𝜓)))
43imp 122 1 ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-in1 577  ax-in2 578
This theorem is referenced by:  pm4.65r  817  dcim  818  imanim  819  pm4.52im  837  exanaliim  1579
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