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Mirrors > Home > ILE Home > Th. List > imanim | GIF version |
Description: Express implication in terms of conjunction. The converse only holds given a decidability condition; see imandc 827. (Contributed by Jim Kingdon, 24-Dec-2017.) |
Ref | Expression |
---|---|
imanim | ⊢ ((𝜑 → 𝜓) → ¬ (𝜑 ∧ ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | annimim 823 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓)) | |
2 | 1 | con2i 595 | 1 ⊢ ((𝜑 → 𝜓) → ¬ (𝜑 ∧ ¬ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-in1 582 ax-in2 583 |
This theorem is referenced by: difdif 3140 ssdif0im 3366 inssdif0im 3369 nominpos 8751 |
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