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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege46 | Structured version Visualization version GIF version |
Description: If 𝜓 holds when 𝜑 occurs as well as when 𝜑 does not occur, then 𝜓 holds. If 𝜓 or 𝜑 occurs and if the occurences of 𝜑 has 𝜓 as a necessary consequence, then 𝜓 takes place. Identical to pm2.6 183. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege46 | ⊢ ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege33 38969 | . 2 ⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) | |
2 | frege45 38982 | . 2 ⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((¬ 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 38923 ax-frege2 38924 ax-frege8 38942 ax-frege28 38963 ax-frege31 38967 ax-frege41 38978 |
This theorem is referenced by: frege47 38984 |
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