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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege47 | Structured version Visualization version GIF version |
Description: Deduce consequence follows from either path implied by a disjunction. If 𝜑, as well as 𝜓 is sufficient condition for 𝜒 and 𝜓 or 𝜑 takes place, then the proposition 𝜒 holds. Proposition 47 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege47 | ⊢ ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜒) → ((𝜑 → 𝜒) → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege46 42210 | . 2 ⊢ ((¬ 𝜑 → 𝜒) → ((𝜑 → 𝜒) → 𝜒)) | |
2 | frege21 42187 | . 2 ⊢ (((¬ 𝜑 → 𝜒) → ((𝜑 → 𝜒) → 𝜒)) → ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜒) → ((𝜑 → 𝜒) → 𝜒)))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜒) → ((𝜑 → 𝜒) → 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-frege1 42150 ax-frege2 42151 ax-frege8 42169 ax-frege28 42190 ax-frege31 42194 ax-frege41 42205 |
This theorem is referenced by: frege48 42212 frege49 42213 |
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