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Mirrors > Home > ILE Home > Th. List > pm2.53 | GIF version |
Description: Theorem *2.53 of [WhiteheadRussell] p. 107. This holds intuitionistically, although its converse does not (see pm2.54dc 877). (Contributed by NM, 3-Jan-2005.) (Revised by NM, 31-Jan-2015.) |
Ref | Expression |
---|---|
pm2.53 | ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.24 611 | . 2 ⊢ (𝜑 → (¬ 𝜑 → 𝜓)) | |
2 | ax-1 6 | . 2 ⊢ (𝜓 → (¬ 𝜑 → 𝜓)) | |
3 | 1, 2 | jaoi 706 | 1 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 605 ax-io 699 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: ori 713 ord 714 orel1 715 pm2.63 790 notnotrdc 829 dfordc 878 pm5.6r 913 xorbin 1366 19.33b2 1609 r19.30dc 2604 onsucelsucexmid 4489 oprabidlem 5852 omnimkv 7099 xnn0nnn0pnf 9166 absle 10989 |
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