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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege45 | Structured version Visualization version GIF version |
Description: Deduce pm2.6 194 from con1 148. Proposition 45 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege45 | ⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege44 41133 | . 2 ⊢ ((¬ 𝜓 → 𝜑) → ((𝜑 → 𝜓) → 𝜓)) | |
2 | frege5 41085 | . 2 ⊢ (((¬ 𝜓 → 𝜑) → ((𝜑 → 𝜓) → 𝜓)) → (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-frege1 41075 ax-frege2 41076 ax-frege8 41094 ax-frege28 41115 ax-frege31 41119 ax-frege41 41130 |
This theorem is referenced by: frege46 41135 |
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