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| Description: Closed form of a syllogism followed by a swap of antecedents. Proposition 18 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| frege18 | ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜑) → (𝜓 → (𝜃 → 𝜒)))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | frege5 43818 | . 2 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜑) → (𝜃 → (𝜓 → 𝜒)))) | |
| 2 | frege16 43834 | . 2 ⊢ (((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜑) → (𝜃 → (𝜓 → 𝜒)))) → ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜑) → (𝜓 → (𝜃 → 𝜒))))) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜑) → (𝜓 → (𝜃 → 𝜒)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 | 
| This theorem was proved from axioms: ax-mp 5 ax-frege1 43808 ax-frege2 43809 ax-frege8 43827 | 
| This theorem is referenced by: frege19 43842 frege23 43843 frege20 43846 frege51 43873 frege64a 43900 frege64b 43927 frege64c 43945 frege82 43963 | 
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