Theorem imanim 784

Description: Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 785. (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 781	. 2 ⊢ ((φ ∧ ¬ ψ) → ¬ (φ → ψ))
2	1	con2i 557	1 ⊢ ((φ → ψ) → ¬ (φ ∧ ¬ ψ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-in1 544  ax-in2 545

This theorem is referenced by:  difdif  3063  npss0  3260  ssdif0im  3280  inssdif0im  3285  nominpos  7939