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| Mirrors > Home > MPE Home > Th. List > Mathboxes > frege9 | Structured version Visualization version GIF version | ||
| Description: Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 43813 only in an unessential way. Identical to imim1 83. Proposition 9 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| frege9 | ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | frege5 43813 | . 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | |
| 2 | ax-frege8 43822 | . 2 ⊢ (((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) → ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 | 
| This theorem was proved from axioms: ax-mp 5 ax-frege1 43803 ax-frege2 43804 ax-frege8 43822 | 
| This theorem is referenced by: frege11 43827 frege10 43833 frege19 43837 frege21 43840 frege37 43853 frege56aid 43883 frege56a 43884 frege61a 43892 frege56b 43911 frege61b 43919 frege56c 43932 frege61c 43937 frege117 43993 frege130 44006 frege132 44008 | 
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