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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 884. (Contributed by Jim Kingdon, 24-Dec-2017.) |
Ref | Expression |
---|---|
imanim | ⊢ ((𝜑 → 𝜓) → ¬ (𝜑 ∧ ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | annimim 681 | . 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓)) | |
2 | 1 | con2i 622 | 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-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-in1 609 ax-in2 610 |
This theorem is referenced by: difdif 3252 ssdif0im 3478 inssdif0im 3481 nominpos 9108 |
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