Theorem imanim 683

Description: Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 884. (Contributed by Jim Kingdon, 24-Dec-2017.)

Assertion

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

Proof of Theorem imanim

Step	Hyp	Ref	Expression
1		annimim 681	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
2	1	con2i 622	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:  difdif  3252  ssdif0im  3479  inssdif0im  3482  nominpos  9115