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Mirrors > Home > ILE Home > Th. List > pm2.65 | GIF version |
Description: Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. Proofs, such as this one, which assume a proposition, here 𝜑, derive a contradiction, and therefore conclude ¬ 𝜑, are valid intuitionistically (and can be called "proof of negation", for example by Section 1.2 of [Bauer] p. 482). By contrast, proofs which assume ¬ 𝜑, derive a contradiction, and conclude 𝜑, such as condandc 866, are only valid for decidable propositions. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.) |
Ref | Expression |
---|---|
pm2.65 | ⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.27 40 | . . . 4 ⊢ (𝜑 → ((𝜑 → ¬ 𝜓) → ¬ 𝜓)) | |
2 | 1 | con2d 613 | . . 3 ⊢ (𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓))) |
3 | 2 | a2i 11 | . 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → ¬ (𝜑 → ¬ 𝜓))) |
4 | 3 | con2d 613 | 1 ⊢ ((𝜑 → 𝜓) → ((𝜑 → ¬ 𝜓) → ¬ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-in1 603 ax-in2 604 |
This theorem is referenced by: pm4.82 934 |
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