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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege5 | Structured version Visualization version GIF version |
Description: A closed form of syl 17. Identical to imim2 58. Theorem *2.05 of [WhiteheadRussell] p. 100. Proposition 5 of [Frege1879] p. 32. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege5 | ⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-frege1 40156 | . 2 ⊢ ((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓))) | |
2 | frege4 40165 | . 2 ⊢ (((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-frege1 40156 ax-frege2 40157 |
This theorem is referenced by: rp-frege25 40171 frege6 40172 frege7 40174 frege9 40178 frege12 40179 frege16 40182 frege25 40183 frege18 40184 frege22 40185 frege14 40189 frege29 40197 frege34 40203 frege45 40215 frege80 40309 frege90 40319 |
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