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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 43790 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 43790 | . 2 ⊢ ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → (𝜑 → 𝜒))) | |
2 | ax-frege8 43799 | . 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 43780 ax-frege2 43781 ax-frege8 43799 |
This theorem is referenced by: frege11 43804 frege10 43810 frege19 43814 frege21 43817 frege37 43830 frege56aid 43860 frege56a 43861 frege61a 43869 frege56b 43888 frege61b 43896 frege56c 43909 frege61c 43914 frege117 43970 frege130 43983 frege132 43985 |
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