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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 892). (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 | pm2.24 622 | . 2 ⊢ (𝜑 → (¬ 𝜑 → 𝜓)) | |
2 | ax-1 6 | . 2 ⊢ (𝜓 → (¬ 𝜑 → 𝜓)) | |
3 | 1, 2 | jaoi 717 | 1 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in2 616 ax-io 710 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: ori 724 ord 725 orel1 726 pm2.63 801 notnotrdc 844 dfordc 893 pm5.6r 928 xorbin 1395 19.33b2 1640 r19.30dc 2637 onsucelsucexmid 4550 oprabidlem 5931 omnimkv 7189 xnn0nnn0pnf 9287 absle 11139 |
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