Theorem imanim 689

Description: Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 890. (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 687	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
2	1	con2i 628	1 ⊢ ((𝜑 → 𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-in1 615  ax-in2 616

This theorem is referenced by:  difdif  3275  ssdif0im  3502  inssdif0im  3505  nominpos  9174