Theorem imanim 699

Description: Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 901. (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 697	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓))
2	1	con2i 636	1 ⊢ ((𝜑 → 𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-in1 623  ax-in2 624

This theorem is used by:  difdif  3354  ssdif0im  3589  inssdif0imOLD  3593  nominpos  9543