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Mirrors > Home > ILE Home > Th. List > annimim | GIF version |
Description: Express conjunction in terms of implication. One direction of Theorem *4.61 of [WhiteheadRussell] p. 120. The converse holds for decidable propositions, as can be seen at annimdc 926. (Contributed by Jim Kingdon, 24-Dec-2017.) |
Ref | Expression |
---|---|
annimim | ⊢ ((𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.27 40 | . . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓)) | |
2 | con3 632 | . . 3 ⊢ (((𝜑 → 𝜓) → 𝜓) → (¬ 𝜓 → ¬ (𝜑 → 𝜓))) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝜑 → (¬ 𝜓 → ¬ (𝜑 → 𝜓))) |
4 | 3 | imp 123 | 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 604 ax-in2 605 |
This theorem is referenced by: pm4.65r 677 imanim 678 pm4.52im 740 dcim 831 exanaliim 1634 |
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