Theorem annimim 681

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 932. (Contributed by Jim Kingdon, 24-Dec-2017.)

Assertion

Ref	Expression
annimim	⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))

Proof of Theorem annimim

Step	Hyp	Ref	Expression
1		pm2.27 40	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
2		con3 637	. . 3 ⊢ (((𝜑 → 𝜓) → 𝜓) → (¬ 𝜓 → ¬ (𝜑 → 𝜓)))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → (¬ 𝜓 → ¬ (𝜑 → 𝜓)))
4	3	imp 123	1 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-in1 609  ax-in2 610

This theorem is referenced by:  pm4.65r  682  imanim  683  pm4.52im  745  dcim  836  exanaliim  1640