Theorem imanim 818

Description: Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 819. (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 815	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
2	1	con2i 589	1 ⊢ ((𝜑 → 𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))

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 576  ax-in2 577

This theorem is referenced by:  difdif  3097  ssdif0im  3308  inssdif0im  3311  nominpos  8268